Cambridge Additional Mathematics

Start at and work

around to , noting down

the angle every time you

reach points A and B.

¡ 2

2

¼

¼

Start at angle and work

around to , noting down

the angle every time you

reach points A and B.

0

3 ¼

-_Qw

O

AUh_p BQ_____QyQ_p

AB-Qw

3 ¼ OO 0

Trigonometric functions (Chapter 9) 245

Example 8 Self Tutor

Solve exactly for 06 x 63 ¼: a sinx=¡^12 b sin 2x=¡^12

The equations both have the form sinμ=¡^12.

There are two points on the unit circle with sine¡^12.

They correspond to angles^76 ¼ and^116 ¼.

a In this caseμis simplyx,sowe

have the domain 06 x 63 ¼.

The only solutions for this domain

are x=^76 ¼ or^116 ¼.

b In this caseμis 2 x.

If 06 x 63 ¼ then 062 x 66 ¼.

) 2 x=^76 ¼,^116 ¼,^196 ¼,^236 ¼,^316 ¼,or^356 ¼

) x=^712 ¼,^1112 ¼,^1912 ¼,^2312 ¼,^3112 ¼,or^3512 ¼

3 Solve exactly for 06 x 63 ¼: a cosx=^12 b cos 2x=^12

4 Solve exactly for 06 x 62 ¼: a sinx=¡p^12 b sin 3x=¡p^12

5 Find the exact solutions of:

a cosx=¡^12 , 06 x 65 ¼ b 2 sinx¡1=0, ¡ 360 ± 6 x 6360 ±

c 2 cosx+

p

3=0, 06 x 63 ¼ d 3 cos 2x+3=0, 06 x 63 ¼

e 4 cos 3x+2=0, ¡¼ 6 x 6 ¼

Example 9 Self Tutor

Solve tan 2x+1=2for ¡¼ 6 x 6 ¼.

tan 2x=1

There are two points on the unit

circle which have tangent 1.

Since ¡¼ 6 x 6 ¼,

¡ 2 ¼ 62 x 62 ¼

So, 2 x=¡^74 ¼, ¡^34 ¼, ¼ 4 ,or^54 ¼

) x=¡^78 ¼, ¡^38 ¼, ¼ 8 ,or^58 ¼

A_rp

BTf_p

O

_rp

-2¼

2 ¼

4037 Cambridge

cyan magenta yellow black Additional Mathematics

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100
Y:\HAESE\CAM4037\CamAdd_09\245CamAdd_09.cdr Friday, 4 April 2014 1:21:28 PM BRIAN