Cambridge Additional Mathematics

a

LESS SPACE GIVEN!!!

LESS SPACE GIVEN!!!

208 The unit circle and radian measure (Chapter 8)

Theunit circleis the circle with

centre(0,0)and radius 1 unit.

CIRCLES WITH CENTRE(0,0)

Consider a circle with centre(0,0)and radiusrunits.

Suppose P(x,y) is any point on this circle.

Since OP=r,

p

(x¡0)^2 +(y¡0)^2 =r fdistance formulag

) x^2 +y^2 =r^2

x^2 +y^2 =r^2 is the equation of a circle with centre(0,0)

and radiusr.

The equation of theunit circleis x^2 +y^2 =1.

ANGLE MEASUREMENT

Suppose P lies anywhere on the unit circle, and A is(1,0).

Letμbe the angle measured from [OA] on the positivex-axis.

μis positivefor anticlockwise rotations and

negativefor clockwise rotations.

For example: μ= 210±=^76 ¼

Á=¡ 150 ±=¡^56 ¼

DEFINITION OF SINE AND COSINE

Consider a point P(a,b) which lies on the unit circle in the first

quadrant. [OP]makes an angleμwith thex-axis as shown.

Using right angled triangle trigonometry:

cosμ=

ADJ

HYP

=

a

1

=a

sinμ=

OPP

HYP

=

b

1

=b

tanμ=

OPP

ADJ

=

b

a

=

sinμ

cosμ

C

TRIGONOMETRIC RATIOS

C The unit circle and the trigonometric ratios

y

x

1

-1 1

-1

O

y

x

q

P

A

1

Positive

direction

Negative

direction

O

b

P,(a b)

1

1

1

μ

x

y

O a

Á

μ

x

y

30 ° O

x

y

r

P,(x y)

O(0 0),

cyan magenta yellow black

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_08\208CamAdd_08.cdr Monday, 6 January 2014 11:42:27 AM BRIAN