Cambridge Additional Mathematics

The unit circle and radian measure (Chapter 8) 211

PERIODICITY OF TRIGONOMETRIC RATIOS

Since there are 2 ¼radians in a full revolution, if we add any integer multiple of 2 ¼toμ(in radians) then

the position of P on the unit circle is unchanged.

Forμin radians and k 2 Z,

cos (μ+2k¼) = cosμ and sin (μ+2k¼) = sinμ.

We notice that for any point (cosμ,sinμ) on the unit circle, the

point directly opposite is (¡cosμ,¡sinμ)

) cos(μ+¼)=¡cosμ

sin(μ+¼)=¡sinμ

and tan(μ+¼)=

¡sinμ

¡cosμ

=

sinμ

cosμ

= tanμ

Forμin radians andk 2 Z, tan(μ+k¼) = tanμ.

Thisperiodicfeature is an important property of the trigonometric functions.

EXERCISE 8C

1 For each unit circle illustrated:

i state the exact coordinates of points A, B, and C in terms of sine and cosine

ii use your calculator to give the coordinates of A, B, and C correct to 3 significant figures.

ab

2 With the aid of a unit circle, complete the following table:

μ(degrees) 0 ± 90 ± 180 ± 270 ± 360 ± 450 ±

μ(radians)

sine

cosine

tangent

y

1 x

1

μ

• 1


• 1

μ¼+

(a b),

(-a -b),

O

y

1 x

1

26 °

146 ° A

B

C

-1

-1 O

199 °

y

x

1

1

• 35 °

123 °

251 °

A

B

C

-1

-1 O

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_08\211CamAdd_08.cdr Monday, 23 December 2013 1:56:36 PM BRIAN