Cambridge Additional Mathematics

214 The unit circle and radian measure (Chapter 8)

EXERCISE 8D.1

1 Find the possible values ofcosμfor:

a sinμ=^12 b sinμ=¡^13 c sinμ=0 d sinμ=¡ 1

2 Find the possible values ofsinμfor:

a cosμ=^45 b cosμ=¡^34 c cosμ=1 d cosμ=0

Example 8 Self Tutor

If sinμ=¡^34 and ¼<μ<^32 ¼, findcosμandtanμ. Give exact answers.

Now cos^2 μ+ sin^2 μ=1

) cos^2 μ+ 169 =1

) cos^2 μ= 167

) cosμ=§

p

7

4

But ¼<μ<^32 ¼,soμis a quadrant 3 angle.

) cosμ is negative.

) cosμ=¡

p

7

4 and tanμ=

sinμ

cosμ

=

¡^34

¡

p

7

4

=p^37

3 Find the exact value of:

a sinμif cosμ=^23 and 0 <μ<¼ 2 b cosμ if sinμ=^25 and ¼ 2 <μ<¼

c cosμ if sinμ=¡^35 and^32 ¼<μ< 2 ¼ d sinμ if cosμ=¡ 135 and ¼<μ<^32 ¼.

4 Find the exact value of tanμ given that:

a sinμ=^13 and ¼ 2 <μ<¼ b cosμ=^15 and^32 ¼<μ< 2 ¼

c sinμ=¡p^13 and ¼<μ<^32 ¼ d cosμ=¡^34 and ¼ 2 <μ<¼.

Example 9 Self Tutor

If tanμ=¡ 2 and^32 ¼<μ< 2 ¼, find sinμ and cosμ. Give exact answers.

tanμ=

sinμ

cosμ

=¡ 2

) sinμ=¡2 cosμ

Now sin^2 μ+ cos^2 μ=1

) (¡2 cosμ)^2 + cos^2 μ=1

) 4 cos^2 μ+ cos^2 μ=1

) 5 cos^2 μ=1

) cosμ=§p^15

But^32 ¼<μ< 2 ¼,soμis a quadrant 4 angle.

) cosμis positive andsinμis negative.

) cosμ=p^15 and sinμ=¡p^25.

y

x

S A

TC

q

• Er

O

y

x

S A

TC

q

O

cyan magenta yellow black

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100 4037 Cambridge
Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_08\214CamAdd_08.cdr Monday, 6 January 2014 11:45:41 AM BRIAN