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iii) Consider ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDQ4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
But
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1OCIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDU4IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNTAiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyNTAgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMzQiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAxMzQgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
Alternative: Using
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDM4IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Dividing by cos3θ numerator and denominator
Applications of the double and triple formulae
A. Proving Identities
Examples: Prove the following identities
(i)
+ ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNzMiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxNzMgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
(ii)
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDQ4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
(iii) ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMTEiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyMTEgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
Solution(i)
I. Proof ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNTAiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyNTAgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
Dealing with L.H.S
=cos2A+cos2A-sin2A
=2cos2A-sin2A
=2cos2A-sin2A
=2 – ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMzQiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxMzQgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=
R.H.S
II. Solution(ii)
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Dealing with L.H.S
But ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNzMiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxNzMgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
III. Solution(iii)
![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMTEiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyMTEgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNTQiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxNTQgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDQ4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Work on the following problems prove the identities
i)
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDQ4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
ii)
iii) ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNzMiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAxNzMgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
iv)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI5NiIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDk2IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
v)
+
=2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI2NyIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDY3IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
vi)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDQ4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
vii)
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDM4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
viii)
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMzQiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxMzQgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
ix)
= 2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDQ4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
x)
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDM4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
xi)
+
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDM4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Warm up with:
i) Find tan
without calculate mathematical tables
ii) ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI3NyIgaGVpZ2h0PSIxOSIgdmlld0JveD0iMCAwIDc3IDE5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
HALF ANGLES FORMULAE
From
=
– ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDQ4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Then ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNTQiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAxNTQgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMTEiIGhlaWdodD0iNDgiIHZpZXdCb3g9IjAgMCAyMTEgNDgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMzQiIGhlaWdodD0iNDgiIHZpZXdCb3g9IjAgMCAxMzQgNDgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=
= 1 – ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDQ4IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxOTIiIGhlaWdodD0iNDgiIHZpZXdCb3g9IjAgMCAxOTIgNDgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=
– 1 +![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI2NyIgaGVpZ2h0PSI0OCIgdmlld0JveD0iMCAwIDY3IDQ4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Again from
=
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But
= 1 –![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI2NyIgaGVpZ2h0PSI0OCIgdmlld0JveD0iMCAwIDY3IDQ4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
2
= 1 –![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDM4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
For
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSI0OCIgdmlld0JveD0iMCAwIDQ4IDQ4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Similarly the formulae can be expressed as
EQUATION OF THE FORM
a
= c
where a, b and c are real constant.
The task here is to solve the equation. The are two ways to solve.
i. Using t –fomulae
ii. Using R – fomula (or transforming a function a
+ b
= c as a single function)
I. USING t- FORMULAE
Consider ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMjUiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxMjUgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
Concept of t formulae From
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI5NiIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDk2IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNDQiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAxNDQgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMTUiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAxMTUgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMDYiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxMDYgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
But
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDM4IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Again
= 1 + ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDQ4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Let
= y
from Pythagoras theorem
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMjEiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyMjEgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMjEiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyMjEgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMzQiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxMzQgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI3NyIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDc3IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNzMiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxNzMgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
Then
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyOSIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDI5IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Cos2ÆŸ =
…………………… (ii)
From equations (i) (ii) and (iii) it follows that
Let t =
, then we get
Equation (1), (2) and (3) are called t-substitution formulae
Solving the equation
Let t =![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDM4IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
a – at² + 2bt = c(1 + t²)
a – at² + 2bt = c + ct²
at² + ct² – 2bt + c – a =o
(a + c)t² – 2bt + c –a =o
Quadratic equation
Solve for it
t= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxOTIiIGhlaWdodD0iNDgiIHZpZXdCb3g9IjAgMCAxOTIgNDgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMjUiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAxMjUgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMjUiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAxMjUgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMDYiIGhlaWdodD0iNDgiIHZpZXdCb3g9IjAgMCAxMDYgNDgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
t = ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMTUiIGhlaWdodD0iNDgiIHZpZXdCb3g9IjAgMCAxMTUgNDgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI5NiIgaGVpZ2h0PSI0OCIgdmlld0JveD0iMCAwIDk2IDQ4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
t = ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI5NiIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDk2IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
but t = ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDM4IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
tan
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI5NiIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDk2IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Example:
Solve for values of θ between 0° and 180° if 2cos θ+ sin θ= 2.5
Solution: let t = tan ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMCIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDEwIDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
2
+ 3
=2.5
Then
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyOSIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDI5IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Sin θ= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDM4IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
2
+ 3
2.5
2 -2t² + 6t =2.5![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI2NyIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDY3IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
2– 2t² + 6t = 2.5 + 2.5t²
2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMTEiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyMTEgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
4– 4t² + 12t = 5 + 5t²
9t² – 12t + 1 = 0
t= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNDQiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAxNDQgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI4NiIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDg2IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
=
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDQ4IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1OCIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDU4IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDM4IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
t =
= 1.244 or t =![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDM4IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
t = 0.00893
case 1:
t =1.244, t= tan
tan
= 1.244
t =1.244, t= tan
case 2:
t = 0.0893
t = 0.0893
Example 2: solve the equation
5
– 2
=2 for
for -1800
x ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1OCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDU4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Using t formula, let t =
5cosx – 2sin x=2
5
=2
5 – 5t² -4t = 2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI2NyIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDY3IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
5 – 5t² – 4t = 2 + 2t²
7t² + 4t -3 =0
7t² + 7t – 3t -3 =0
7t (t + 1) -3(t + 1) =0
(7t – 3) (t + 1)=0
7t – 3 = 0 or t + 1=0
7t =3 t= -1
t = ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyOSIgaGVpZ2h0PSI0OCIgdmlld0JveD0iMCAwIDI5IDQ4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Case 1.
t=
= 0.42857
t=
Case2,
t=–1, tan
= â»1
t=–1, tan
II. SOLVING THE EQUATION
acosθ
= C
R-formula or simply transforming a function acosÆŸ
bsinÆŸ as a single function.
From acosθ
bsinθ = c
Consider acosθ + bsinθ – this can be expressed transformed into form
R is the maximum value of a function (or Amplitude)
Then from acosθ + bsinθ =C
acosθ + bsinθ = Rcos(θ –
)
acosθ + bsinθ= R![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMDIiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyMDIgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
Square equation (i) and (ii) then sum
(Rcos
+
= a² + b²
R²cos²
+ R²sin²
= a² + b²
R²
= a² + b²
But
+
=1
R².1 = a² + b²
R² =a² + b²
Then from
acosÆŸ + bsinÆŸ =c = Rcos (ÆŸ – ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxOSIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDE5IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Rcos(![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDQ4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Example
Rcos
cosx = 3cosx
Rcos
= 3 —- (i)
-4sinx = Rsinxsin![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDEwIDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Sin
= 4 —– (ii)
Dividing (ii) by (i), then we get
Dividing (ii) by (i), then we get
= 9 + 16
R²
+ R
= 25
R²1 =25
R= 25, R=
R=5
But ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMzQiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxMzQgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
5![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMTUiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxMTUgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
C = 1.5 ,
= 53.12°
5
= 1.5
Cos
=0.3
X + 53,12°= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI4NiIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDg2IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
X + 53.12° = 72.54°
X = 72.54° – 53.12°
Example 2: solve for
between 0° and 180° if
2
= 2.5
Solution
2
= 2.5
R
=2
3![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDM4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
R![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzNjUiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAzNjUgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
R
=2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDM4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
R
=2 —(i) and
R![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNjMiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxNjMgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
R
= 3 ………. (ii)
Dividing (ii) by (i)
Squaring (i) and (ii) then add
R²
+ R
= 4 + 9
R²![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxOTIiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxOTIgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
R² = 13, R=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyOSIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDI5IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Then ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNDQiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxNDQgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
θ- 56.3°= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDQ4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDM4IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
θ= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNTQiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAxNTQgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=
+ 56.3°
θ= 313.9° + 56.3°= 370.2°
= 370.2° – 360°=10.2°
Example:
3
solve for x iƒ5
– 2sinx =R
=2
3
solve for x iƒ5
5
– 2
= R![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMTEiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyMTEgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
5
= R![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI4NiIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDg2IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
R
= 5 ……………………. (i)
2
= R![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDM4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDM4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
R
= 2 ……………………..(ii)
Dividing (ii) by (i)
Squaring equations (i) and (ii) the add
R²
= 29
R²x1 =29, R²=29, R = ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyOSIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDI5IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
From R
= 2
X + 21.8 = ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI4NiIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDg2IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
X + 21.8° = 68.2° , -68.2°
X= 68.2° – 21.8° = 46.40°
Also x + 21.8° = â»68.2°
X = â»68.2° -21.80 =-90
NB: The R- formula ( Transformation) can also be done using an auxiliary angle approach; where we substitute constants a and b as functions of sine or cosine.
Thus considering the same problem solving 5
– 2
=2
Imagine a triangle
Using Pythagoras theorem
From the figure above, it follows that
Then from 5cos x – 2sin x = 2
So, the principle angle = 21.8°
![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNTAiIGhlaWdodD0iMjMwIiB2aWV3Qm94PSIwIDAgMjUwIDIzMCI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgc3R5bGU9ImZpbGw6I2NmZDRkYjtmaWxsLW9wYWNpdHk6IDAuMTsiLz48L3N2Zz4=)
Using the general solution of sin
X =
– 21.8°
X =
–![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxODIiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxODIgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
X= 68.2° – ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxODIiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxODIgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
n= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMTUiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxMTUgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
find x values according to the limits given in the question
OR imagine a triangle
Then sin
, 2=
sin![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDEwIDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
cos
=
, 5=
cos ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDEwIDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
from 5cosx – 2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI3NyIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDc3IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Using the general solution of cosine
X =
–![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDEwIDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
X=
– 21.8![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDEwIDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
n =![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNTQiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxNTQgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
OTHER KIND OF QUESTIONS USING THE TRANSFORMING INTO A SINGLE FUNCTION CONCEPT
Example:1 Express
i) 4cosx – 5sinx in the form of Rcos(x + ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxOSIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDE5IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
ii) 2sinx + 5cosx in the form of Rsin(x + ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxOSIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDE5IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Solution(i)
4cos x-5sinx =Rcos(x + ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxOSIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDE5IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
4cosx = Rcos
cosx
Rcos
= 4 ……… (i)
5sinx = Rsin
sinx
Rsin
=5 …………..(ii)
Dividing (ii)by (i)
Squaring equations (i) and (ii) then add
R²cos
+ R²
= 16 + 25
R
= 41
R=41, R=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyOSIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDI5IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Rcos
sinx = 2sinx
Rcos
=2 …………(i) and
Rcosxsin
= 5cosx
Rsin
= 5 ………….(ii)
Dividing (ii) by (i)
Tan
= ,
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI5NiIgaGVpZ2h0PSI1OCIgdmlld0JveD0iMCAwIDk2IDU4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Squaring equations (i) and (ii) then add
R²cos²
+ R² sin²
= 4 + 25
R
=29
But cos²![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMjUiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxMjUgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
R²(1)=29
Example. Find the maximum value of 24sinx -7cosx and the smallest positive value of x that gives this maximum value.
Solution. 24sin x -7cosx = Rsin(x – ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxOSIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDE5IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
24sinx = Rcos
sinx
Rcos
=24, 7cosx = Rsin
cosx
Rsin
=7 ………(ii)
Squaring equation (i) and (ii) then add
R
=625
R
=625
R²=625, R=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDM4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
R =25
24
– 7cosx = Rsin![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1OCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDU4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNzMiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxNzMgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=25sin ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI5NiIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDk2IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
24sinx – 7cosx = 25sin![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI5NiIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDk2IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
f(x)= 25sin(x – 16.26°)
Max value of sine function is when
Sin![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMzQiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxMzQgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
X – 16.26°=90°
X = 90° + 16.26°
X= 106.26°
Hence max value f
=y=25 sin 90°
=25
Note. The maximum values of
Problems to work on
Using t formula and R –formula solve the following.
3. 6sinx + 8cosx =6
4. Express 7cosθ+ 24 sinθ in the form of Rcos(10 –![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxOSIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDE5IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
5. Solve for θ
3cosθ + 4sinθ =2
6. 5cos2θ– sin 2θ=2
Note: If the question has no limits/boundaries write the answer using the general solution
FACTOR FORMULAE (SUM AND DIFFERENCE FORMULAE)
The concept here is to express the sum or difference of sine and cosine functions as product and vice versa
Refer
Sin(A +B) = sin AcosB + cosAsin B ……….(i)
Sin(A –B) = sinAcosB –cosAsinB ………….(ii)
Cos(A + B) =cosAcosB – sinA sinB …………(iii)
Cos(A+ B) =cosAcosB + sinAsinB ……………(iv)
Add (i) and (ii)
Sin(A + B) + sin(A +B) =2sin AcosB
Let f = A + B ………(i)
Q =A-B …….(ii)
(a) +(b) 2A = P+Q, A= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyOSIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDI5IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
(a) –(b) 2B =P-Q, B=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyOSIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDI5IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Therefore sin(A+B)+sin(A-B)=2sinAcosBbecome
SinP + sinQ= 2sin
cos
…(1)
Substract(i) –(ii)
Sin(A+B) –sin(A-B) = 2cosA sinB
But P=A+B, Q=A-B
Add (iii) and (iv)
Cos(A+B)+cos(A-B) = 2cosAcosB
CosP + cosQ = 2cos
cos![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDQ4IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Substract (iii) – (iv)
Cos(A + B) –cos(A-B) = -2sinAsin B
Expressions (1) (2) (3) and ( 4) are called factor formulae
APPLICATIONS OF THE FACTOR FORMULAE
a) Proving problems
Examples
i)
= cot 2x
ii)
= cot ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDQ4IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
iii)
= tan![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDQ4IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
v) If A, B and C are angles of a triangle prove that
cosA +cosB + cosC -1 = 4sin
sin
sin ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyOSIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDI5IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
vi) If A, B and C are angles of a triangle prove that
cos2A + cos2B + cos2C + 1 = 4cosAcosBcosC
vii)
=tan A
viii)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDQ4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Solution (i)
(L.H.S)
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNDQiIGhlaWdodD0iNTgiIHZpZXdCb3g9IjAgMCAxNDQgNTgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI4NiIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDg2IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
But
–![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDM4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI5NiIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDk2IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
=
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDM4IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDQ4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDQ4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Solution(ii)
,
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMjUiIGhlaWdodD0iNTgiIHZpZXdCb3g9IjAgMCAxMjUgNTgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1OCIgaGVpZ2h0PSI1OCIgdmlld0JveD0iMCAwIDU4IDU4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMTUiIGhlaWdodD0iNDgiIHZpZXdCb3g9IjAgMCAxMTUgNDgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
Solution (iii)
=
R.H.S
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI3NyIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDc3IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Solution(iv)
= 4![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNTQiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxNTQgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNDQiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAxNDQgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=2cos2Acos![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDM4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
=2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI5NiIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDk2IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
=2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNDQiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAxNDQgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMDYiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxMDYgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDM4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
=2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI5NiIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDk2IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Then
=2
+ 2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI5NiIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDk2IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
=2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxODIiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxODIgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNTAiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAyNTAgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxOTIiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAxOTIgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxOTIiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxOTIgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNzMiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxNzMgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=4
R.H.S
Solution(V).
A, B, C are angles of a![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDEwIDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
A, B, C are angles of a
L.H.S
CosA + cosB + cosC – 1
![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMDIiIGhlaWdodD0iMTQ0IiB2aWV3Qm94PSIwIDAgMjAyIDE0NCI+PHJlY3Qgd2lkdGg9IjEwMCUiIGhlaWdodD0iMTAwJSIgc3R5bGU9ImZpbGw6I2NmZDRkYjtmaWxsLW9wYWNpdHk6IDAuMTsiLz48L3N2Zz4=)
2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzNDYiIGhlaWdodD0iNTgiIHZpZXdCb3g9IjAgMCAzNDYgNTgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=2
-2
………….(i)
But A + B + C= 180°
(Degree angle in ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxOSIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDE5IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
A + B = 180°-C
90 –
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyOSIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDI5IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Apply cos
cos
= cos![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDQ4IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Cos
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1OCIgaGVpZ2h0PSI0OCIgdmlld0JveD0iMCAwIDU4IDQ4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNTAiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAyNTAgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
But ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMjEiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyMjEgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=1 –
–![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1OCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDU4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
= 1 – 2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1OCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDU4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Substitute (ii) into (i)
=2
cos
-2sin![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDM4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
= 2
-2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI2NyIgaGVpZ2h0PSI0OCIgdmlld0JveD0iMCAwIDY3IDQ4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
=2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMTEiIGhlaWdodD0iNTgiIHZpZXdCb3g9IjAgMCAyMTEgNTgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNTAiIGhlaWdodD0iNDgiIHZpZXdCb3g9IjAgMCAyNTAgNDgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
But
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI3NyIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDc3IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Using factor formula
2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzMDciIGhlaWdodD0iNTgiIHZpZXdCb3g9IjAgMCAzMDcgNTgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyOTgiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAyOTggMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxODIiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAxODIgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNzMiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAxNzMgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
But ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMjUiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAxMjUgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxODIiIGhlaWdodD0iNTgiIHZpZXdCb3g9IjAgMCAxODIgNTgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNzMiIGhlaWdodD0iNTgiIHZpZXdCb3g9IjAgMCAxNzMgNTgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNzMiIGhlaWdodD0iNDgiIHZpZXdCb3g9IjAgMCAxNzMgNDgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=4![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNzMiIGhlaWdodD0iNDgiIHZpZXdCb3g9IjAgMCAxNzMgNDgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
solution(VI).
![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMDIiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyMDIgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
= 4![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMzQiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxMzQgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
From factor fomulae
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMDIiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyMDIgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyODgiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAyODggMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyOTgiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyOTggMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
But A + B +C = 180° (
)
A +B = 180° -C
Cos![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMDIiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyMDIgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=
+ ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMDYiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxMDYgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
= –
+ 0
Substitute into (i)
=-2
+
+ 1
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNTQiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxNTQgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI3NyIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDc3IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
= -2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNDAiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyNDAgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
= -2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNjkiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyNjkgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
= -2
+2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDQ4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
=2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMjEiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyMjEgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
But
= –![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI4NiIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDg2IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNTAiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyNTAgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
= -2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNTAiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyNTAgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
= -2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyOTgiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAyOTggMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
= -2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMTEiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAyMTEgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
= -2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNzMiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxNzMgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
= -4![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMzQiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxMzQgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNTQiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxNTQgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI4NiIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDg2IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNTQiIGhlaWdodD0iNTgiIHZpZXdCb3g9IjAgMCAxNTQgNTgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
Solution (vi)
L.H.S changing the products into sin or difference
Numerator: ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI5NiIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDk2IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
From sinP +sinQ=2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNTQiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAxNTQgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI5NiIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDk2IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Similarly
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNDQiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAxNDQgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
Denominator
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI4NiIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDg2IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNTQiIGhlaWdodD0iNTgiIHZpZXdCb3g9IjAgMCAxNTQgNTgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIzOCIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDM4IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
=
RHS
Examples (i) solve for x if
ii) ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxODIiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxODIgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
For ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMzQiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxMzQgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
iii) ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMzAiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyMzAgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
For ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMjUiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxMjUgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
Solution (i)
Writing using factor formulae
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=2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMjUiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxMjUgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
=2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMDYiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxMDYgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxNzMiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxNzMgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMzAiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyMzAgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
2
=1
3x =
=0°, 180, 360°
X=
540°
=0°,60°,120°, 180°
X= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI5NiIgaGVpZ2h0PSIzOCIgdmlld0JveD0iMCAwIDk2IDM4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
X=30°, 150°
iv)
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2
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2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMzQiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAxMzQgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
2
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2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxOTIiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxOTIgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
2x=
2
=1
2x=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNTAiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyNTAgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
X=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMTEiIGhlaWdodD0iMzgiIHZpZXdCb3g9IjAgMCAyMTEgMzgiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
X=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxOTIiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxOTIgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
X= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI5NiIgaGVpZ2h0PSI1OCIgdmlld0JveD0iMCAwIDk2IDU4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
X=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI4NiIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDg2IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
Questions
1. Solve for the value of x between 0° and 360° in the question
i)
–
= ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyOSIgaGVpZ2h0PSI0OCIgdmlld0JveD0iMCAwIDI5IDQ4Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
ii)
+
=0
2. Prove that
i)
+
°=0
ii)
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1OCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDU4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
3. Simplify ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyMDIiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyMDIgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
4. Evaluate ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIxMTUiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAxMTUgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
5. Prove that
2
=![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI0OCIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDQ4IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
If
+
a and
7. Prove that
8. Express as a sum or difference
i) 2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI4NiIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDg2IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
ii) ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI3NyIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDc3IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
iii)
θ
iv) 2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI5NiIgaGVpZ2h0PSIyOSIgdmlld0JveD0iMCAwIDk2IDI5Ij48cmVjdCB3aWR0aD0iMTAwJSIgaGVpZ2h0PSIxMDAlIiBzdHlsZT0iZmlsbDojY2ZkNGRiO2ZpbGwtb3BhY2l0eTogMC4xOyIvPjwvc3ZnPg==)
9. Show without using tables or calculators
i) ![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyNjkiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyNjkgMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
ii) 2![EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2) EcoleBooks | MATHEMATICS As LEVEL(FORM FIVE) NOTES - TRIGONOMETRY(2)](data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSIyOTgiIGhlaWdodD0iMjkiIHZpZXdCb3g9IjAgMCAyOTggMjkiPjxyZWN0IHdpZHRoPSIxMDAlIiBoZWlnaHQ9IjEwMCUiIHN0eWxlPSJmaWxsOiNjZmQ0ZGI7ZmlsbC1vcGFjaXR5OiAwLjE7Ii8+PC9zdmc+)
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