{"id":2845,"date":"2023-10-03T14:01:42","date_gmt":"2023-10-03T14:01:42","guid":{"rendered":"http:\/\/localhost\/ecole9ja\/?p=2845"},"modified":"2023-10-03T14:03:22","modified_gmt":"2023-10-03T14:03:22","slug":"week-8-ss2-first-term-further-mathematics-notes","status":"publish","type":"post","link":"https:\/\/ecolebooks.com\/nigeria\/posts\/week-8-ss2-first-term-further-mathematics-notes\/","title":{"rendered":"Week 8 – SS2 First Term Further Mathematics Notes"},"content":{"rendered":"

\u00a0
\n\u00a0WEEK 8
\n<\/strong>TOPIC: Trigonometric functions
\n<\/strong>The basic trigonometric ratios can be defined in two ways:
\n(i)traditional definition;
\n(ii) modern definition.<\/p>\n

\u00a0Traditional Definition
\n<\/strong>The basic trigonometric ratios can be defined in terms of the sides of a right \u2013 angled triangle.
\n\"\"\/\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Q<\/p>\n

\u00a0
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0r\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0p<\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 P\u00a0\u00a0\u00a0\u00a0 \u0473\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0q\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 R<\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Fig. 14.3
\n<\/strong>\u2206PQR in Fig 14.3 in a right angled triangle with QPR = \u0473 and PRQ = 90\u25e6. We define the three basic rations as follows:
\nCosine of angle \u0473 =
\nSine of angle \u0473 =
\nTangent of \u0473 =
\nThe cosine of angle \u0473, sine of angle \u0473 and the tangent of angle \u0473 will be abbreviated cos\u0473, sin\u0473 and tan\u0473 respectively.
\nThus:
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Cos\u0473 =
\nSin\u0473 =
\nTan\u0473 \u2013
\nAlso:
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 =
\ntan\u0473 = <\/p>\n

\u00a0Ratios of the General Angle
\n<\/strong>First Quadrant
\n<\/strong>\"\"\/
\n\t\t\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 P1<\/sub><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0y<\/p>\n

\u00a0
\n\u00a0
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
\nFig 14.7
\n<\/strong>
\n\u00a0In Fig. 14.7 \u2206OP1<\/sub>N1<\/sub> is a right \u2013 angled triangle constructed from a unit circle.<\/p>\n

\u00a0OP1<\/sub>\u00a0\u00a0\u00a0\u00a0= 1
\nP1N1<\/sub>\u00a0\u00a0\u00a0\u00a0= y
\nON1<\/sub>\u00a0\u00a0\u00a0\u00a0= x
\nP1<\/sub>ON1<\/sub>=\u04731
\nSin\u04731<\/sub>\u00a0\u00a0\u00a0\u00a0= y
\nCos\u0334\u04731<\/sub>\u00a0\u00a0\u00a0\u00a0= x
\nTan\u04731<\/sub>\u00a0\u00a0\u00a0\u00a0=
\nSecond Quadrant
\n<\/strong>\"\"\/
\n\t\tP2
\n\u00a0\u00a0\u00a0\u00a0 1
\ny
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(a)<\/p>\n

\u00a0
\n\u00a0
\n\u00a0
\n\u00a0
\n\u00a0
\n\u00a0\"\"\/\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0y<\/p>\n

\u00a0 P2<\/sub>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0P1<\/sub><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0 \u04721<\/sub>\u00a0\u00a0\u00a0\u00a0 \u04732<\/sub> \u04731<\/sub>
\n\t\t\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x<\/p>\n

\u00a0
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(b)<\/p>\n

\u00a0
\n\u00a0\u00a0\u00a0\u00a0\u00a0Fig. 14.8
\n<\/strong>
\n\u00a0
\n\u00a0From Fig. 14.8(a)
\n\u00a0\u00a0\u00a0\u00a0Sin\u04732<\/sub>\u00a0\u00a0\u00a0\u00a0= y
\n\u00a0\u00a0\u00a0\u00a0Cos\u04732<\/sub>\u00a0\u00a0\u00a0\u00a0= -x
\n\u00a0\u00a0\u00a0\u00a0Tan\u04732<\/sub>\u00a0\u00a0\u00a0\u00a0=
\n = <\/p>\n

\u00a0From Fig. 14.8(b)
\n\u00a0\u00a0\u00a0\u00a0\u04721<\/sub> + \u04732<\/sub>\u00a0\u00a0\u00a0\u00a0= 180\u25e6
\n\u00a0\u00a0\u00a0\u00a0\u04722<\/sub>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0= 180\u25e6 – \u04731<\/sub>
\n\t\t: Sin\u04732<\/sub> = sin(180\u25e6 – \u04731<\/sub>) = y = sin\u04731<\/sub>
\n\t\t\u00a0\u00a0\u00a0\u00a0: Sin(180\u25e6 – \u0473) = sin\u0473
\n: Cos\u04732<\/sub> = cos(180\u25e6 – \u04731<\/sub>) = -x = -cos\u04731<\/sub>
\n\t\t\u00a0\u00a0\u00a0\u00a0:Cos(180\u25e6 – \u0473)\u00a0\u00a0\u00a0\u00a0 = -cos\u0473<\/p>\n

\u00a0Similarly,
\n<\/strong>Tan\u04732<\/sub> = tan (180\u25e6 – \u04731<\/sub>) = 1<\/sub>
\n\t\t\u00a0\u00a0\u00a0\u00a0: tan (180\u25e6 – \u0473) = -tan\u0473
\nHence in the second quadrant:
\nSin(180\u25e6 – \u0473) = sin\u0473
\nCos(180\u25e6 – \u0473) = -cos\u0473
\nTan(180\u25e6 – \u0473) = -tan\u0473<\/p>\n

\u00a0
\n\u00a0
\n\u00a0
\n\u00a0
\n\u00a0
\n\u00a0
\n\u00a0Third Quadrant
\n<\/strong>\"\"\/\"\"\/<\/p>\n

\u00a0
\n\u00a0\u00a0\u00a0\u00a0\u00a0
\n\"\"\/\u00a0\u00a0\u00a0\u00a0 N1<\/sub>\u2013 x\u00a0\u00a0\u00a0\u00a0 \u04733<\/sub>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(a)
\n\"\"\/\"\"\/\"\"\/\u00a0\u00a0\u00a0\u00a0 -y\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Q<\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0P3<\/sub>\u00a0\u00a0\u00a0\u00a01<\/p>\n

\u00a0
\n\u00a0
\n\u00a0
\n\u00a0
\n\u00a0\"\"\/\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0y
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0P1<\/sub>
\n\t\t(b)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u04731<\/sub>
\n\t\t\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u04721<\/sub>
\n\t\t\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u04721<\/sub><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0 P3<\/sub><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Fig. 14.9<\/strong><\/p>\n

\u00a0
\n\u00a0
\n\u00a0Sin\u04733<\/sub>\u00a0\u00a0\u00a0\u00a0= -y
\nCos\u04733<\/sub>\u00a0\u00a0\u00a0\u00a0= -x
\nTan\u04733<\/sub>\u00a0\u00a0\u00a0\u00a0= \u00a0\u00a0\u00a0\u00a0sin\u04723<\/sub> =
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0cos\u04723
\nFrom Fig. 14.9(b)
\n\u04721<\/sub> = 180\u25e6 + \u04721<\/sub>
\n\t\tSIn\u04733<\/sub> = sin(180\u25e6 + \u0472) = -y = -sin\u04721<\/sub>
\n\t\t:sin(180\u25e6 + \u0472) = -sin\u0472
\nCos\u04733<\/sub> = cos(180\u25e6 + \u0472) = -x = -cos\u04721<\/sub>
\n\t\t: cos(180\u25e6 + \u0472) = -sin\u0472<\/p>\n

\u00a0Similarly,
\nTan\u04733<\/sub>\u00a0\u00a0\u00a0\u00a0= \u00a0\u00a0\u00a0\u00a0tan (180\u25e6 + \u04721<\/sub>)= = tan\u04731<\/sub>
\n\t\t\u00a0\u00a0\u00a0\u00a0: tan(180\u25e6 + \u0473) = tan\u0473
\nHence in the third quadrant:<\/p>\n

\u00a0Sin(180\u25e6 + \u0473) = -sin\u0473
\nCos(180\u25e6 + \u0473) = -cos\u0473
\nTan(180\u25e6 + \u0473) = tan\u0473<\/p>\n

\u00a0
\n\u00a0Fourth Quadrant
\n<\/strong>(a)
\n\"\"\/<\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x<\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 -y
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u04721<\/sub>\u00a0\u00a0\u00a0\u00a01
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 P4
\n<\/sub>
\n\u00a0
\n\u00a0
\n\u00a0\"\"\/\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0y\u00a0\u00a0\u00a0\u00a0<\/p>\n

\u00a0(b)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 P1<\/sub>
\n\t\t\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u04721<\/sub>
\n\t\t\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 \u04721<\/sub>\u00a0\u00a0\u00a0\u00a0 x
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u04721<\/sub>
\n\t\t\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 P2<\/sub><\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Fig. 14.10<\/strong>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
\n\t\t\t<\/strong>Sin\u04734<\/sub> = y
\nCos\u04734<\/sub> = x
\nTan\u04734<\/sub> =
\nFrom Fig, 14.10(b)
\n\u04724<\/sub> + \u04721<\/sub>= 360\u25e6
\n\u00a0\u00a0\u00a0\u00a0\u04724<\/sub> = 360\u25e6 – \u0473
\nSIn\u04731<\/sub> = sin(360\u25e6 + \u0472) = -y = -sin\u04721<\/sub>
\n\t\t: sin(360\u25e6 + \u0472) = -sin\u0472
\nCos\u04731<\/sub> = cos(360\u25e6 + \u0472) = -x = -cos\u04721<\/sub>
\n\t\t: cos(360\u25e6 + \u0472) = -sin\u0472
\nTan\u04734<\/sub>\u00a0\u00a0\u00a0\u00a0= \u00a0\u00a0\u00a0\u00a0tan (360\u25e6 + \u0472)= = -tan\u0473
\n\u00a0\u00a0\u00a0\u00a0: tan(360\u25e6 + \u0473) = -tan\u0473
\nHence in the third quadrant:
\nSin(360\u25e6 + \u0473) = -sin\u0473
\nCos(360\u25e6 + \u0473) = cos\u0473
\nTan(360\u25e6 + \u0473) = -tan\u0473
\n(a) In the first quadrant, all the ratios are positive.
\n(b) In the second quadrant, only sin ratio is positive, while the rest are negative.
\n(c) In the third quadrant, only tangent ratio is positive, while the rest are negative.
\n(d) In the fourth quadrant, only cosine ratio is positive, while the rest are negative.<\/p>\n

\u00a0These observations can be summarized in the figure below:
\n\"\"\/\"\"\/\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0y\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0y<\/p>\n

\u00a0 SINL\u00a0\u00a0\u00a0\u00a0ALL\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0S\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0A
\nT\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0C\u00a0\u00a0\u00a0\u00a0<\/p>\n

\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(a)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 (b)<\/p>\n

\u00a0
\n\u00a0
\n\u00a0
\n\u00a0
\n\u00a0
\n\u00a0
\n\u00a0Negative Angles
\n<\/strong>\"\"\/\"\"\/\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0y<\/p>\n

\u00a0
\n\u00a0
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0360\u25e6 – 0
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x<\/p>\n

\u00a0
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 P\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Ps
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(a)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(b)
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Fig. 14.12<\/p>\n

\u00a0
\n\u00a0Since negative angles are measured in the clockwise sense, the direction of OP when rotated through \u2013\u0473 is the same as where it is rotated through 360\u25e6 – \u0473.
\nHence in the forth quadrant:
\nSin(-\u0473) sin sin(360\u25e6 – \u0473) = -sin\u0473
\nCos(-\u0473) cos(360\u25e6 – \u0473) = -cos\u0473
\nTan(-\u0473) tan(360\u25e6 – \u0473) = -tan\u0473<\/p>\n

\u00a0Use tables to evaluate each of the following:
\n<\/strong>(a) sin 143\u25e6\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(b) cos 115\u25e6
\n(c) tan 125\u25e6<\/p>\n

\u00a0Solution
\n<\/strong>(a) 143\u25e6 is in the second quadrant, so
\nSin143\u25e6 = sin(180\u25e6 – 143\u25e6)
\n\u00a0\u00a0\u00a0\u00a0= sin37\u25e6
\n\u00a0\u00a0\u00a0\u00a0= 0.6018\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
\n(b) 115\u25e6 is in the second quadrant, so
\nCos115\u25e6 = -cos(180\u25e6 – 115\u25e6)
\n\u00a0\u00a0\u00a0\u00a0= -cos65\u25e6
\n\u00a0\u00a0\u00a0\u00a0= -0.4226
\n\u00a0\u00a0\u00a0\u00a0
\n(c) 125\u25e6 is in the second quadrant, so<\/p>\n

\u00a0Tan125\u25e6 = -tan(180\u25e6 – 125\u25e6)
\n\u00a0\u00a0\u00a0\u00a0= -tan55\u25e6
\n\u00a0\u00a0\u00a0\u00a0= -1.428<\/p>\n

\u00a0Use tables to evaluate each of the following
\n(a) sin230\u25e6\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(b) cos236\u25e6
\n(c)tan 242\u25e6<\/p>\n

\u00a0Solution<\/strong>
\n\t\t220\u25e6, 236\u25e6 and 242\u25e6 are all in the third quadrant, hence;
\n(a) sin220\u25e6 = sin(180\u25e6 + 40\u25e6)
\n= -sin40\u25e6
\n= -0.6428<\/p>\n

\u00a0(b) cos236\u25e6 = cos(180\u25e6 + 56\u25e6)
\n\u00a0\u00a0\u00a0\u00a0= -cos56\u25e6
\n\u00a0\u00a0\u00a0\u00a0= -0.5992<\/p>\n

\u00a0(c) tan242\u25e6 = tan(180\u25e6 + 62\u25e6)
\n\u00a0\u00a0\u00a0\u00a0= tan62\u25e6
\n\u00a0\u00a0\u00a0\u00a0= 1.881<\/p>\n

\u00a0Use tables to evaluate each of the following:
\n(a) sin310\u25e6\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(b) cos285\u25e6
\n(c) 334\u25e6<\/p>\n

\u00a0Solution
\n<\/strong>310\u25e6, 285\u25e6 and 334\u25e6 are all in the fourth quadrant, hence;
\n(a) sin310\u25e6 = sin(360\u25e6 – 50\u25e6)
\n\u00a0\u00a0\u00a0\u00a0= -sin50\u25e6
\n\u00a0\u00a0\u00a0\u00a0= -0.7660<\/p>\n

\u00a0(b) cos285\u25e6 = cos(360\u25e6 – 75\u25e6)
\n\u00a0\u00a0\u00a0\u00a0= cos75\u25e6
\n\u00a0\u00a0\u00a0\u00a0= 0.2588<\/p>\n

\u00a0(c) tan334\u25e6 = tan(360\u25e6 – 26\u25e6)
\n\u00a0\u00a0\u00a0\u00a0= -tan26\u25e6
\n\u00a0\u00a0\u00a0\u00a0= -0.4877
\nUse tables to evaluate each of the following
\n(a) cos(-30\u25e6)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(b) sin(-60\u25e6)
\n(c) tan(-120\u25e6)<\/p>\n

\u00a0Solution
\n<\/strong>(a) cos(-30\u25e6) = cos330\u25e6
\n\u00a0\u00a0\u00a0\u00a0= cos30\u25e6
\n\u00a0\u00a0\u00a0\u00a0= 0.8660<\/p>\n

\u00a0(b) sin(-60\u25e6) = sin300\u25e6
\n\u00a0\u00a0\u00a0\u00a0= -sin60\u25e6
\n\u00a0\u00a0\u00a0\u00a0= -8660<\/p>\n

\u00a0(c) tan(-120\u25e6) = tan240\u25e6
\n\u00a0\u00a0\u00a0\u00a0= tan60\u25e6
\n\u00a0\u00a0\u00a0\u00a0= 1.732<\/p>\n

\u00a0Use the table to find the value of \u0473 between \u0473\u25e6 and 360\u25e6 which satisfy each of the following:
\n(a) cos\u0473 = -0.4540
\n(b) tan\u0473 = 1.176
\n(c) sin\u0473 = -0.9336<\/p>\n

\u00a0Solution
\n<\/strong>(a) The cosine ratio is negative in the second and third quadrants. First find the acute angle whose cosine is 0.4540
\nFrom the tables cos 63\u25e6 = 0.4540
\n: In the second quadrant
\n\u00a0\u00a0\u00a0\u00a0\u0472 = 180\u25e6 – 63\u25e6
\n\u00a0\u00a0\u00a0\u00a0= 117\u25e6
\nIn the third quadrant,
\n\u00a0\u00a0\u00a0\u00a0\u0472 = 180\u25e6 + 63\u25e6
\n\u00a0\u00a0\u00a0\u00a0= 243\u25e6<\/p>\n

\u00a0(b) The tangent ratio is positive in the first and third quadrants.
\nFirst find the acute angle whose tangent is 1.176.
\nFrom the tables.
\nTan49.62\u25e6= 1.176\u25e6
\nIn the first quadrant.
\n\u0472 = 49.62\u25e6
\nIn the third quadrant.
\n\u00a0\u00a0\u00a0\u00a0\u0472 = 180\u25e6+ 49.62\u25e6
\n\u00a0\u00a0\u00a0\u00a0= 229.62\u25e6<\/p>\n

\u00a0(c) The sine ratio is negative in the third and fourth quadrant.
\nFirst find the acute angles whose sine ratio is 0.9336.
\nFrom tables.
\nSin69\u25e6 = 0.9336
\nIn the third quadrant
\n\u00a0\u00a0\u00a0\u00a0\u0472 = 180\u25e6 + 69\u25e6
\n\u00a0\u00a0\u00a0\u00a0= 249\u25e6
\nIn the fourth quadrant.
\n\u00a0\u00a0\u00a0\u00a0\u0472 = 360\u25e6 – 69\u25e6
\n\u00a0\u00a0\u00a0\u00a0= 291\u25e6<\/p>\n

\u00a0Evaluation
\n<\/strong><\/p>\n

    \n
  1. In what quadrant are the followings ; tan ( -540) , cos (- 1080)\n<\/li>\n<\/ol>\n

    \u00a0General Evaluation
    \n<\/strong>(1) Prove that \u00a0\u00a0\u00a0\u00a0(1 \u2013sin\u0473) (1 + sin\u0473)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0= cot2<\/sup>\u0473
    \n\"\"\/\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0sin2<\/sup>\u0473\u00a0\u00a0\u00a0\u00a0
    \n(2) Show that (sec\u0473 \u2013 tan\u0473) (sec\u0473 + tan\u0473) = 1
    \n(3) Find the values of sin (-210) in surd form<\/p>\n

    \u00a0Reading Assignment<\/strong>
    \n\t\tF\/Maths Project 1 pages 225 \u2013 247 Exercise 14 Q1, 3, 5, 7 and 8<\/p>\n

    \u00a0Weekend Assignment
    \n<\/strong>Given that sin\u0473 =
    \n(1)Find cos\u0473\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(a)
    \n(2) Find tan\u0473\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
    \n(3) Find cosec\u0473\u00a0\u00a0\u00a0\u00a0
    \n(4) Find sec\u0473\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(a)
    \n(5) Find cot\u0473\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/p>\n

    \u00a0THEORY
    \n<\/strong>1) Prove that 1\/1+cosx + 1\/1-cosx = 2 cosec2<\/sup> x
    \n2) Given that sin x = 5\/13 and x is acute find cosec x
    \n , cot x and sec x<\/p>\n","protected":false},"excerpt":{"rendered":"

    \u00a0 \u00a0WEEK 8 TOPIC: Trigonometric functions The basic trigonometric ratios can be defined in two…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,230],"tags":[],"class_list":["post-2845","post","type-post","status-publish","format-standard","hentry","category-posts","category-first-term-ss2-further-mathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/2845","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/comments?post=2845"}],"version-history":[{"count":1,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/2845\/revisions"}],"predecessor-version":[{"id":2846,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/2845\/revisions\/2846"}],"wp:attachment":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/media?parent=2845"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/categories?post=2845"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/tags?post=2845"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}