Cambridge Additional Mathematics

218 The unit circle and radian measure (Chapter 8)

MULTIPLES OF¼ 6 OR 30 ±

Since OA=OP, triangle OAP is isosceles.

The remaining angles are therefore also 60 ±, and so

triangle AOP is equilateral.

The altitude [PN] bisects base [OA], so ON=^12.

If P is(^12 ,k), then (^12 )^2 +k^2 =1 fPythagorasg

) k^2 =^34

) k=

p

3

2 fas k>^0 g

So, P is (^12 ,

p

3

2 ) where

p

3

2 ¼^0 :^866.

cos¼ 3 =^12 and sin¼ 3 =

p

3

2

Now NbPO=¼ 6 =30±.

Hence cos¼ 6 =

p

3

2 and sin

¼

6 =

1

2

You should remember these values. If you forget, divide in two an

equilateral triangle with side length 2.

Formultiples of¼ 6 , we have:

Summary

² Formultiples of¼ 2 , the coordinates of the points on the unit circle involve 0 and§ 1.

² Forothermultiples of¼ 4 , the coordinates involve§p^12.

² Forothermultiples of¼ 6 , the coordinates involve§^12 and§

p

3

2.

² The signs of the coordinates are determined by which quadrant the angle is in.

You should be able to use this summary to find the trigonometric ratios for angles which are multiples of

¼

6 and

¼

4.

30 °

60 °

1

P

O Qw N

]

30 °

60 °

22

1

2

~` 3

x

y

A,() 10

1

N

P,(_ k)Qw_

Qw

60 °

1

k

O

x

y

() 01 ,¡

() 10 ,¡

( -1) 0 ,

(-1 ),¡ 0

&Qw]_, *

&],Qw*

&],-Qw*

&-w]Q,_- * &Qw]_-, *

&-],-Qw*

&-],Qw*

&-Qw]_, *

Wd_p

Uh_p

O

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_08\218CamAdd_08.cdr Monday, 6 January 2014 9:40:28 AM BRIAN