Cambridge Additional Mathematics

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GRAPHING

PACKAGE

Make sure you find

the solutions on

the given domain.

all

x

12345678

y

2

3

-1

-2

-3

1

y=tanx

11 22 33 44 55 66 77 88 xx

yy

22

33

-1-1

-2-2

-3-3

11

y=y=tantanxx

OO

Trigonometric functions (Chapter 9) 243

4

The graph of y= tanx is illustrated.

a Use the graph to estimate: i tan 1 ii tan 2: 3

Check your answers using a

b Find, correct to 1 decimal place, the solutions of:

i tanx=2for 06 x 68 ii tanx=¡ 1 : 4 for 26 x 67.

5 Use the graphing package to solve forxon the domain 0 <x< 4 ¼:

a sinx=0: 431 b cosx=¡ 0 : 814 c 3 tanx¡2=0

6 Use the graphing package to solve forxon the domain ¡ 56 x 65 :

a 5 cosx¡4=0 b 2 tanx+13=0 c 8 sinx+3=0

7aUse the graphing package to solve sin^2 x+ sinx¡2=0

for 06 x 62 ¼.

b Solve form: m^2 +m¡2=0.

c Hence explain your answer ina.

SOLVING TRIGONOMETRIC EQUATIONS USING ALGEBRA

Using a graph we get approximate decimal ornumericalsolutions to trigonometric equations.

Sometimes exact solutions are needed in terms of¼, and these arise when the solutions are multiples of

¼

6 or

¼

4. Exact solutions obtained using algebra are calledanalyticalsolutions.

We use the periodicity of the trigonometric functions to give us all solutions in the required domain.

For example, consider sinx=1. We know from the unit circle that a

solution is x=¼ 2. However, since the period of sinx is 2 ¼, there

are infinitely many solutions spaced 2 ¼ apart.

Hence x=¼ 2 +k 2 ¼ is a solution for any k 2 Z.

In this course we will be solving equations on a fixed domain. This means

there will be a finite number of solutions.

calculator.

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_09\243CamAdd_09.cdr Tuesday, 28 January 2014 9:42:11 AM BRIAN