Cambridge Additional Mathematics

The unit circle and radian measure (Chapter 8) 209

In general, for a point P anywhere on the unit circle:

² cosμis thex-coordinate of P

² sinμ is they-coordinate of P

We can hence find the coordinates of any point on the unit circle with given angleμmeasured from the

positivex-axis.

For example:

Since the unit circle has equation x^2 +y^2 =1, (cosμ)^2 + (sinμ)^2 =1 for allμ.

We commonly write this as cos^2 μ+ sin^2 μ=1.

For all points on the unit circle, ¡ 16 x 61 and ¡ 16 y 61.

So, ¡ 16 cosμ 61 and ¡ 16 sinμ 61 for allμ.

DEFINITION OF TANGENT

Suppose we extend [OP] to meet the tangent from A(1,0).

We let the intersection between these lines be point Q.

Note that as P moves, so does Q.

The position of Q relative to A is defined as the tangent

function.

Notice that 4 s ONP and OAQ are equiangular and therefore

similar.

Consequently AQ

OA

=NP

ON

and hence AQ

1

=sinμ

cosμ

.

Under the definition that AQ= tanμ, tanμ=

sinμ

cosμ

.

y

1 x

1

327 °

O -33°

( 327 327 )

( (-33 ) (-33 ))

cos °,sin ° or

°, °

__

cos__sin

x

y

A,() 10 ¡

tan¡μ

Q,(1¡¡tanμ)

P

μ

cos¡μ N

1

1

-1

-1 tangent

O

sin¡μ

y

1 x

1 ()cos75 sin75°°,

()cos165 sin165°°,

()cos255 sin255°°,

75 °

165 °

255 °

O

x

y

q

1

-1 1

-1

P,()cos sin¡q¡q

x O

y

q

1

-1 1

-1

P,()cos sin¡q¡q

O

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_08\209CamAdd_08.cdr Friday, 4 April 2014 11:56:39 AM BRIAN