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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

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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.

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  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

0.50

0.87

1.00

0.87

0.50

0

-0.50

-0.87

-0.50

-0.87

-0.50

0

3 sin x

0

1.50

2.61

3.00

2.61

1.50

0

-1.50

-2.61

-1.50

-2.61

-1.50

0

 

 

 

 

 

390

420

450

480

510

540

570

600

630

660

690

720

0.5

0.87

1.00

0.87

0.50

0

-0.50

-0.87

-1.00

-0.87

-0.50

0

1.50

2.61

3.00

2.61

1.50

0

-1.50

-2.61

-3.00

-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 .

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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.

 

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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

0.5000

      

-08660

    

y

0

1.00

      

-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

00

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

3600

Cos x0

1.00

 

0.50

  

-0.87

 

-0.87

     

2 cos ½ x0

2.00

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)

0.50

  

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

00

200

400

600

800

1000

1200

1400

1600

1800

-3cos 2x0

-3.00

-2.30

-0.52

1.50

2.82

2.82

1.50

-0.52

-2.30

-3.00

2 sin (3 x0 + 300)

1.00

1.73

2.00

1.73

1.00

0.00

-1.00

-1.73

-2.00

-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

00

300

600

90

10

1500

180

210

240

270

300

330

360

Cosx0

1.00

 

0.50

  

-0.87

 

-0.87

     

2cos ½ x0

2.00

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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EcoleBooks | Mathematics Form 1-4 : CHAPTER FIFTY NINE - TRIGONOMETRY III

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