Cambridge Additional Mathematics

The unit circle and radian measure (Chapter 8) 215

5 Find exact values forsinxandcosxgiven that:

a tanx=^23 and 0 <x<¼ 2 b tanx=¡^43 and ¼ 2 <x<¼

c tanx=

p

5

3 and ¼<x<

3 ¼

2 d tanx=¡

12

5 and

3 ¼

2 <x<^2 ¼

6 Suppose tanμ=k wherekis a constant and ¼<μ<^32 ¼. Write expressions for sinμ and

cosμ in terms ofk.

FINDING ANGLES WITH PARTICULAR TRIGONOMETRIC RATIOS

FromExercise 8Cyou should have discovered that:

Forμin radians:

² sin(¼¡μ) = sinμ ² cos(¼¡μ)=¡cosμ ² cos(2¼¡μ) = cosμ

We need results such as these, and also the periodicity of the trigonometric ratios, to find angles which have

a particular sine, cosine, or tangent.

Example 10 Self Tutor

Find the two anglesμon the unit circle, with 06 μ 62 ¼, such that:

a cosμ=^13 b sinμ=^34 c tanμ=2

a cos¡^1 (^13 )¼ 1 : 23

) μ¼ 1 : 23 or 2 ¼¡ 1 : 23

) μ¼ 1 : 23 or 5 : 05

b sin¡^1 (^34 )¼ 0 : 848

) μ¼ 0 : 848 or ¼¡ 0 : 848

) μ¼ 0 : 848 or 2 : 29

c tan¡^1 (2)¼ 1 : 11

) μ¼ 1 : 11 or ¼+1: 11

) μ¼ 1 : 11 or 4 : 25

EXERCISE 8D.2

1 Find two anglesμon the unit circle, with 06 μ 62 ¼, such that:

a tanμ=4 b cosμ=0: 83 c sinμ=^35

d cosμ=0 e tanμ=1: 2 f cosμ=0: 7816

g sinμ= 111 h tanμ=20: 2 i sinμ=^3940

If , , or is positive,

your calculator will give

in the domain.

cos sin tan

0

μμ μ

μ

<μ<_wp

-1 1 x

1

y

-1

-1 1 x O

1

y

-1

Er

-1 1 x O

1 y

-1 Qe

O

4037 Cambridge

cyan magenta yellow black Additional Mathematics

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100
Y:\HAESE\CAM4037\CamAdd_08\215CamAdd_08.cdr Friday, 4 April 2014 11:59:54 AM BRIAN