Cambridge Additional Mathematics

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230 Trigonometric functions (Chapter 9)

In previous studies of trigonometry we have only considered static situations where an angle is fixed.

However, when an object moves around a circle, the situation is dynamic. The angleμbetween the radius

[OP] and the positivex-axis continually changes with time.

Consider again theOpening Problemin which a Ferris wheel of radius

10 m revolves at constant speed. We let P represent the green light on

the wheel.

The height of P relative to thex-axis can be determined using right angled

triangle trigonometry:

sinμ=

h

10

,soh= 10 sinμ.

As time goes by,μchanges and so doesh.

So, we can writehas a function ofμ, or alternatively we can writehas a function of timet.

For example, suppose the Ferris wheel observed by Andrew takes 100 seconds for a full revolution. The

graph below shows the height of the light above or below the principal axis against the time in seconds.

We observe that the amplitude is 10 metres and the period is 100 seconds.

THE BASIC SINE CURVE y= sinx

Suppose point P moves around the unit circle so the angle [OP] makes

with the positive horizontal axis isx. In this case P has coordinates

(cosx,sinx).

If we project the values ofsinxfrom the unit circle to a set of axes

alongside, we can obtain the graph of y= sinx.

Note carefully thatxon the unit circle diagram is anangle, and becomes

the horizontal coordinate of the sine function.

Unless indicated otherwise, you should assume thatxis measured in radians. Degrees are only included on

this graph for the sake of completeness.

B The sine function

DEMO

10

-10

height metres()

O 5050 100100 time (seconds)

x

P,(x x)cos sin

O

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xx

yy

OO

11

-1-1

_wppw_ p s_E_s_E_pp 22 pp

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y=y=sinsinxx

h

μ

10

P

10 x

y

O

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_09\230CamAdd_09.cdr Monday, 6 January 2014 4:31:08 PM BRIAN