Cambridge Additional Mathematics

The unit circle and radian measure (Chapter 8) 213

b In which quadrants are the following true?

i cosμis positive. ii cosμis negative.

iii cosμandsinμare both negative. iv cosμis negative andsinμis positive.

8aIf AOPb =BbOQ=μ, what is the measure of AOQ?b

b Copy and complete:

[OQ] is a reflection of [OP] in the ......

and so Q has coordinates ......

c What trigonometric formulae can be deduced

fromaandb?

9aCopy and complete:

μc sinμ sin(¡μ) cosμ cos(¡μ)

0 : 75

1 : 772

3 : 414

6 : 25

¡ 1 : 17

b What trigonometric formulae can be deduced from your results ina?

c The coordinates of P in the figure are (cosμ,sinμ).

i By finding the coordinates of Q in terms ofμintwo

differentways, prove your formulae inb.

ii Hence explain why cos(2¼¡μ) = cosμ.

The identity cos^2 μ+ sin^2 μ=1 is essential for finding trigonometric ratios.

Example 7 Self Tutor

Find the possible values of cosμ for sinμ=^23. Illustrate your answers.

cos^2 μ+ sin^2 μ=1

) cos^2 μ+

¡ 2

3

¢ 2

=1

) cos^2 μ=^59

) cosμ=§

p

5

3

D Applications of the unit circle

x

1

A

-1 1

B

1

y

P,()cos sinqq

q q

Q

O

-1

-1 1 x

1

y

P

q

• q

Q

O

-1

y

x

We

1

O 1

p 5

• 3

p 5

3

4037 Cambridge

cyan magenta yellow black Additional Mathematics

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100
Y:\HAESE\CAM4037\CamAdd_08\213CamAdd_08.cdr Tuesday, 21 January 2014 2:45:22 PM BRIAN