Trigonometric ratios 3 Questions and Answers

Trigonometric ratios 3 Questions

1.  Complete the table below by filling in the blank spaces.

 (2mks)

On the grid provided, using a scale of 1 cm to represent 30⁰ on the horizontal axis and 4cm to represent 1 unit on the vertical axis draw the graph of y = cos x⁰ and y = 2 cos ½ x⁰.  (4mks)

 (a) State the period and amplitude of y = 2 cos ½ x⁰ (2mks)

 (b) Use your graph to solve the equation 2 cos ½ x – cos x = 0.  (2mks)

2.  a) Complete the table below by filling in the blank spaces

1. On the grid provided, draw on the same set of axes the graphs of and for. Using a scale o 1 cm for 15⁰ on axis and 2 cm for I unit on the y-axis   (5mks)
2. State the period of (1mk)
3. Solve the equation (2mks)

3.  Complete the table below by filling in the blank spaces.

 Using the scale 1 cm to represent 30^o on the horizontal axis and 4cm to represent 1 unit on the vertical axis, draw on the grid provided, the graph of y = cos x° and y = 2 cos ½ x°

 a)  Find the period and amplitude of y = 2cos ½ x° (2mks)

b)  Describe the transformation that maps the graph of y = Cos x° on the graph of

y = 2 cos ½ x°.  (2mks)

4.  The table below gives some values of y = sin 2x and y = 2 cox is the range given.

 (a) Complete

 (b) On the same axes, draw the graphs of y = sin 2x and y = 2 cos x.

 (c) Use your graph to find in values of x for which sin 2x – 2 cos x = 0.

 (d) From your graph

  (i) Find the highest point of graph y = sin 2x.

  (ii) The lowest point of graph y = 2 cos x.  

5.  (a) Copy and complete the table below for y =2sin (x +15)^o and y =cos(2x -30)^o for 0^o
 x  360^o

(b) On the same axis draw the graphs:

y = 2sin (x + 15) and y = cos(2x -30) for  0^o
 x  360^o  

(c) Use your graph to:

(i) State the amplitudes of the functions y = 2sin (x +15) and y= cos (2x -30)

(ii) Solve the equation 2sin (x+15) – cos (2x -30) = 0

6.  The diagram below shows a frustum of a square based pyramid. The base ABCD is a

square of side 10cm. The top A¹B¹C¹D¹ is a square of side 4cm and each of the slant edges

of the frustum is 5cm

Determine the:

i) Altitude of the frustrum

ii) Angle between AC1 and the base ABCD

 iii) Calculate the volume of the frustrum  

7.  (a) Compete the table below:  

 y = 3sin (2x + 15) ^o  

 (b) Use the table to draw the curve y = 3sin (2x +15) for the values – 180^o

120^o  

 (c) Use the graph to find:

  (i) The amplitude  

  (ii) The period  

  (iii) The solution to the equation:-

  Sin (2x + 15)^o = ¹/₃

8.  Make q the subject of the formula in A

B

9.  a) Complete the table below for the functions y = cos (2x + 45)^o and y = -sin (x + 30^o)for

– 180^o ≤ x ≤ 180^o.  

 b) On the same axis, draw the graphs of y = cos (2x + 45)^o and y = -sin (x + 30)^o

c) Use the graphs drawn in (b) above to solve the equation.

Cos (2x + 45)^o + sin(x + 30)^o = 0

10.  Without using tables or calculators evaluate sin 60^o cos 60^o leaving your answer in surd form. tan 30^o sin 45^o  

11.  (a) Complete the table below for the functions y = 3 sin x and y = 2 cos x

(b) Using a scale of 2cm to represent 1 unit on the y- axis and 1cm to present 30^o on the

x-axis ,draw the graphs of y =3sinx and y = 2cosx on the same axes on the grid provided

 (c) From your graphs:

  (i) State the amplitude of y = 3sin x  

(ii) Find the values of x for which 3sin x – 2cos x = 0  

  (iii) Find the range of values of x for which 3sin x  2cos x

12.  (a) Fill in the following table of the given function:-

(b) On the grid provided draw the graph of the function y = sin ½x
and y = 3Sin (½x + 60)

on the same set of axes

 (c) What transformation would map the function y = sin ½
x
onto y = 3 Sin (½ x + 60)

 (d) (i) State the period and amplitude of function : y = 3 Sin (½x + 60  

  (ii) Use your graph to solve the equation: 3Sin ( ½x + 60) – Sin ½x = 0    

13.  a) Complete the table below giving your answer to 2 decimal places  

b) On the grid provided, using the same axis and scale draw the graphs of :-

y = 2sinxº, and y =1-cosx for

   0º≤ x ≤ 180º , take the scale of

 2cm for 30º on the x-axis

 2cm for 1 unit on the y-axis  

 c) use the graph in (b)above too solve the equation 2sinx + cosxº = 1 and determine the

range of values of for which 2 sinxº =1-cosxº

14.  Solve the equation 2 sin (x + 30) = 1 for 0 ≤ x ≤ 360.

15.  (a) Complete the table below, giving your values correct to 1 decimal place

(b) Draw a graph of y = 10 sin x for values of x from 0^o to 180^o. Take the scale 2cm represents

20^o on the x-axis and 1cm represents 1 unit on the y axis

(c) By drawing a suitable straight line on the same axis, solve the equation: –

500 sin x = -x + 250

16.  Complete the table below for the functions y = cosx and y =2 cos (x 300) for ≤x ≤ 3600

(a) On the same axis, draw the graphs of y cos x and y 2cos(x – 30) for O

 (b) (i) State the amplitude of the graph y = cos x^o.

  (ii) State the period of the graph y = 2 cos (x + 30°).

 c) Use your graph to solve

Cos x = 2cos(x+30°)

17.  Solve the equation sin(2 +10) = -0.5  

 for 0 

 2^c  

18.  Solve the equation

4 sin 2x = 5 – 4 cos²
x for 0° ≤ x ≤ 360°

19.  (a) Complete the table given below by filling in the blank spaces  

 (b) On the grid provided; draw on the same axes, the graphs of y = 4cos 2x and

  y =2sin(2x +30^o)
for 0^o  _X 
180^o. Take the scale: 1cm for 15^o on the x-axis and

  2cm for 1unit on the y-axis    

 (c) From your graph:-

  (i) State the amplitude of y = cos 2x  

  (ii) Find the period of y = 2sin (2x + 30^o)

 (d) Use your graph to solve:-

4cos2x – 2sin (2x +30) = 0

Trigonometric ratios 3 Answers

1.  a)

b)

(c) -90^o or 90^o

(d) (i) Highest point 1 unit

  Lowest point – 1.4

2.

(i) Amplitudes:, y = 2 sin ( x + 15)

  = 2units

  y = cos (2x – 30)

  =1unit

12^o, 159^o

3.  Determine the

i) Altitude of the frustrum

 Solution

A¹C¹ = √ 4² + 4² = √32 = 4√2

AC = √10² + 10²

= √200

= 10√2

AM + XM = 10√2 – 4√2

 = 6√2

AM = ^6√2/₂ = 3√2

Height = AM =√ 5² – (3√ 2)² = √25 – 18

 = √7 = 2.646

  the altitude of the frustrum = 2.646 cm

ii) Angle between AC and the base ABCD

AX = 3√2 + 4√2 = 7√2

Tan ø = ^CX/_AX = ^√7/ _7√2 = ^2.646/_9.898

= 0.2673

 = tan ^-10.2673

= 14.96°

iii) Volume of pyramid = ¹/₃ bh

AC = 10 √ 2

A1C1= 4 √ 2

L.S.F = 10:4


h + 2.646 = 10

h 4

4(h + 2.646) = 10h

4h + 10.584 = 10h

6h = 10.584

h = 1.764

H = h + 2.646

= 1.764 + 2.646 = 4.410

Vf = (¹/₃ x 10 x 10 x 4.41) – (¹/₃ x 4 x 4 x 1.76)

= ^441.0/₃ – ^28.224/₃

= ^413.776/₃

= 137.592cm³

4.  (a) table completed

(b)

(c) (i) 3 P1 – plotting

  S1- scale

  C1 – smooth curve

 (ii) 180^o

 (iii) Line y = 1 drawn

  x = 4.5^o or 72.8^{o –} 107.2^o – 175.4^o

5.  (^A/_B)² = p + 33q

q – 3P

A²q – 3A²P = BP + 3Bq

Aq² – 3Bq = BP + 3A²P

2(A² – 3B) = BP + 3A²P

  Q = BP + 3A²P

  A² – 3B

6.

7. 7.  3 x ½

  2

  1 x 1

  3 2

  3 x 6

  4 1

 18

4

3
2

 4

8. a)

(c) (i) Amplitude =3

  (ii) x = 36^o

  x = 216^o

(iii) 33^o
 x  213^o

9.

10.

11.  Sin (x + 30) = 0.5

 x + 30 = 30^o

 x = 0

 0, 180, 360

12.   (c)10sin x = ^-1/₅₀ + 5

Y = ^-1/₅₀ + 5

X₁ = 28^o
1

X₂ = 70^o
1

12.    

b)  i) amplitude = 1

ii) Period = 360°

iii) 45°, 219°

13.  2 + 10 = 210^o, 330^o, 570^o, 690^o

2 = 200, 320, 560, 680

= 100^o, 160^o, 280^o, 340^o

= 5^c
, 8^c, 14^c ,17^c

  90 9 9 9

14.  4sin 2x+4cos x-5 = 0

4(1-cos2X) + 4 cosx -5= 0

4cos2x – 4 cosx + 1=0

4cos2x – 2cosx – 2cos x +1 =0

(2cos x – 1)2 = 0

X = 60°, 300°

15.  

(b) graph

(c)(i) Amplitude = 4

  (ii) period = 180^o

 (d) x = 30^o, 120^o