Specific Objectives

By the end of the topic the learner should be able to:

(a) Recall and define trigonometric ratios;

(b) Derive trigonometric identity sin2x+cos2x = 1;

(c) Draw graphs of trigonometric functions;

(d) Solve simple trigonometric equations analytically and graphically;

(e) Deduce from the graph amplitude, period, wavelength and phase angles.

Content

(a) Trigonometric ratios

(b) Deriving the relation sin2x+cos2x =1

(c) Graphs of trigonometric functions of the form

y = sin x y = cos x, y = tan x

y = a sin x, y = a cos x,

y = a tan x y = a sin bx,

y = a cos bx y = a tan bx

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y = a sin(bx ± 9)

y = a cos(bx ± 9)

y = a tan(bx ± 9)

(d) Simple trigonometric equation

(e) Amplitude, period, wavelength and phase angle of trigonometric functions.

Introduction

Consider the right – angled triangle OAB

AB = r

OA = r

Since triangle OAB is right- angled

Divide both sides by gives

Example

If tanshow that;

Solution

Factorize the numerator gives and since

But

Therefore, =

Example

Show that

Removing the brackets from the expression gives

Using

Also

Therefore

Example

Given that

1.
2.
3.

Solution using the right angle triangle below.

1. cos

therefore =

2. =
3. =1

Waves

Amplitude

This is the maximum displacement of the wave above or below the x axis.

Period

The interval after which the wave repeats itself

Transformations of waves

The graphs of y = sin x and y = 3 sin x can be drawn on the same axis. The table below gives the corresponding values of sin x and 3 sin x for

 0 30 60 90 120 150 180 210 240 270 300 330 360 Sin x 0 0.5 0.87 1 0.87 0.5 0 -0.5 -0.87 -0.5 -0.87 -0.5 0 3 sin x 0 1.5 2.61 3 2.61 1.5 0 -1.5 -2.61 -1.5 -2.61 -1.5 0

 390 420 450 480 510 540 570 600 630 660 690 720 0.5 0.87 1 0.87 0.5 0 -0.5 -0.87 -1 -0.87 -0.5 0 1.5 2.61 3 2.61 1.5 0 -1.5 -2.61 -3 -2.61 -2.61 0

The wave of y = 3 sin x can be obtained directly from the graph of y = sin x by applying a stretch scale factor 3 , x axis invariant .

Note;

• The amplitude of y= 3sinx is y =3 which is three times that of y = sin x which is y =1.
• The period of the both the graphs is the same that is or 2

Example

Draw the waves y = cos x and y = cos . We obtain y = cos from the graph y = cos x by applying a stretch of factor 2 with y axis invariant.

Note;

• The amplitude of the two waves are the same.
• The period of y = cos is that is, twice the period of y = cos x

Trigonometric Equations

In trigonometric equations, there are an infinite number of roots. We therefore specify the range of values for which the roots of a trigonometric equation are required.

Example

Solve the following trigonometric equations:

1. Sin 2x = cos x, for
2. Tan 3x = 2, for
3.

Solution

1. Sin 2 x = cos x

Sin 2x = sin (90 – x)

Therefore 2 x = 90 – x

X =

For the given range, x =.

2. Tan 3x = 2

From calculator

3x =.

In the given range;

3.

Sin sin

End of topic

 Did you understand everything?If not ask a teacher, friends or anybody and make sure you understand before going to sleep!

Past KCSE Questions on the topic

1.  (a)  Complete the table for the function y = 2 sin x

 x 00 100 200 300 400 500 600 700 800 900 1000 1100 1200 Sin 3x 0 0.5 -8660 y 0 1 -1.73

(b)  (i)  Using the values in the completed table, draw the graph of

y = 2 sin 3x for 00 ≤ x ≤ 1200 on the grid provided

(ii)  Hence solve the equation 2 sin 3x = -1.5

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

 X0 0 300 600 900 1200 1500 1800 2100 2400 2700 3000 3300 3600 Cos x0 1 0.50 -0.87 -0.87 2 cos ½ x0 2 1.93 0.52 -1.00 -2.00

Using the scale 1 cm to represent 300 on the horizontal axis and 4 cm to represent 1 unit on the vertical axis draw, on the grid provided, the graphs of y = cosx0 and y = 2 cos ½ x0 on the same axis.

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

(b)  Describe the transformation that maps the graph of y = cos x0 on the graph of y = 2 cos 1/2 x0

1. (a)  Complete the table below for the value of y = 2 sin x + cos x.
 X 00 300 450 600 900 1200 1350 1500 1800 2250 2700 3150 3600 2 sin x 0 1.4 1.7 2 1.7 1.4 1 0 -2 -1.4 0 Cos x 1 0.7 0.5 0 -0.5 -0.7 -0.9 -1 0 0.7 1 Y 1 2.1 2.2 2 1.2 0.7 0.1 -1 -2 -0.7 1

(b)  Using the grid provided draw the graph of y=2sin x + cos x for 00. Take 1cm represent 300 on the x- axis and 2 cm to represent 1 unit on the axis.

(c)  Use the graph to find the range of x that satisfy the inequalities

2 sin x cos x > 0.5

4.  (a)  Complete the table below, giving your values correct to 2 decimal places.

 x 0 10 20 30 40 50 60 70 Tan x 0 2 x + 300 30 50 70 90 110 130 150 170 Sin ( 2x + 300) 0.5 1

b)  On the grid provided, draw the graphs of y = tan x and y = sin ( 2x + 300) for 00 ≤ x 700

Take scale:  2 cm for 100 on the x- axis

4 cm for unit on the y- axis

Use your graph to solve the equation tan x- sin ( 2x + 300 ) = 0.

5.  (a)  Complete the table below, giving your values correct to 2 decimal places

 X0 0 30 60 90 120 150 180 2 sin x0 0 1 2 1 1 – cos x0 0.5 1

(b)  On the grid provided, using the same scale and axes, draw the graphs of

y = sin x0 and y = 1 – cos x0 ≤ x ≤ 1800

Take the scale: 2 cm for 300 on the x- axis

2 cm for I unit on the y- axis

(c)  Use the graph in (b) above to

(i)  Solve equation

2 sin xo + cos x0 = 1

1. Determine the range of values x for which 2 sin xo> 1 – cos x0

6.  (a)  Given that y = 8 sin 2x – 6 cos x, complete the table below for the missing

values of y, correct to 1 decimal place.

 X 00 150 300 450 600 750 900 1050 1200 Y = 8 sin 2x – 6 cos x -6 -1.8 3.8 3.9 2.4 0 -3.9

(b)  On the grid provided, below, draw the graph of y = 8 sin 2x – 6 cos for

00 ≤ x ≤ 1200

Take the scale 2 cm for 150 on the x- axis

2 cm for 2 units on the y – axis

(c)  Use the graph to estimate

(i)  The maximum value of y

(ii)  The value of x for which 4 sin 2x – 3 cos x =1

7.  Solve the equation 4 sin (x + 300) = 2 for 0 ≤ x ≤ 3600

8.  Find all the positive angles not greater than 1800 which satisfy the equation

Sin2 x – 2 tan x = 0

Cos x

9.  Solve for values of x in the range 00 ≤ x ≤ 3600 if 3 cos2 x – 7 cos x = 6

10.  Simplify 9 – y2 where y = 3 cos θ

y

11.  Find all the values of Ø between 00 and 3600 satisfying the equation 5 sin Ө = -4

12.  Given that sin (90 – x) = 0.8. Where x is an acute angle, find without using mathematical tables the value of tan x0

13.  Complete the table given below for the functions

y= -3 cos 2x0 and y = 2 sin (3x/20 + 30) for 0 ≤ x ≤ 1800

 X0 0 200 400 600 800 1000 1200 1400 1600 1800 -3cos 2x0 -3 -2.3 -0.52 1.5 2.82 2.82 1.5 -0.52 -2.3 -3 2 sin (3 x0 + 300) 1 1.73 2 1.73 1 0 -1 -1.73 -2 -1.73

Using the graph paper draw the graphs of y = -3 cos 2x0 and y = 2 sin (3x/20 + 300)

(a)  On the same axis. Take 2 cm to represent 200 on the x- axis and 2 cm to represent one unit on the y – axis

(b)  From your graphs. Find the roots of 3 cos 2 x0 + 2 sin (3x/20 + 300) = 0

14.  Solve the values of x in the range 00 ≤ x ≤ 3600 if 3 cos2x – 7cos x = 6

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

 x0 0 300 600 90 10 1500 180 210 240 270 300 330 360 Cosx0 1 0.50 -0.87 -0.87 2cos ½ x0 2 1.93 0.5

Using the scale 1 cm to represent 300 on the horizontal axis and 4 cm to represent 1 unit on the vertical axis draw on the grid provided, the graphs of y – cos x0 and y = 2 cos ½ x0 on the same axis

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

Ans. Period = 7200. Amplitude = 2

1. Describe the transformation that maps the graph of y = cos x0 on the graph of y = 2 cos ½ x0

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