Cambridge Additional Mathematics

The unit circle and radian measure (Chapter 8) 219

Example 12 Self Tutor

Find the exact values of sin®, cos®, and tan® for:

a ®=^34 ¼ b ®=^43 ¼

a^34 ¼is a multiple of¼ 4.

The angle lies in quadrant 2 , so only

sin^34 ¼ is positive.

sin(^34 ¼)=p^12

cos(^34 ¼)=¡p^12

tan(^34 ¼)=¡ 1

b^43 ¼is a multiple of¼ 6.

The angle lies in quadrant 3 , so only

tan^43 ¼ is positive.

sin(^43 ¼)=¡

p 3

2

cos(^43 ¼)=¡^12

tan(^43 ¼)=

¡

p

3

2

¡^12

=

p

3

EXERCISE 8E

1 Use a unit circle diagram to find exact values for sinμ, cosμ, and tanμ, forμequal to:

a ¼ 4 bc^74 ¼ d ¼ e ¡ 43 ¼

2 Use a unit circle diagram to find exact values for sin ̄, cos ̄, and tan ̄, for ̄equal to:

a ¼ 6 b^23 ¼ c^76 ¼ d^53 ¼ e^116 ¼

3 Find the exact values of:

a cos 120±, sin 120±, and tan 120± b cos(¡ 45 ±), sin(¡ 45 ±), and tan(¡ 45 ±)

4a

b

Example 13 Self Tutor

Without using a calculator, show that 8 sin(¼ 3 ) cos(^56 ¼)=¡ 6.

sin(¼ 3 )=

p 3

2 and cos(

5 ¼

6 )=¡

p 3

2

) 8 sin(¼ 3 ) cos(^56 ¼)=8(

p

3

2 )(¡

p

3

2 )

=2(¡3)

=¡ 6

x

y

¼0.71

¼¡-0.71

O

&-[[, *

Ef_p

x

y

¼¡-0.87

O

Rd_p

&-Qw],_- *

• Qw

x

y

O

&Qw],_ *

&-],Qw*

Th_p e_p

¼

2

Find the exact values ofcos 270±andsin 270±.

What can you say abouttan 270±?

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_08\219CamAdd_08.cdr Friday, 24 January 2014 3:22:15 PM BRIAN