Cambridge Additional Mathematics

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Trigonometric functions (Chapter 9) 231

x

Click on the icon to generate the sine function for yourself.

You should observe that the sine function can be continued beyond 06 x 62 ¼

in either direction.

The unit circle repeats itself after one full revolution, so theperiodof y= sinx is 2 ¼.

Themaximumvalue is 1 and theminimumis¡ 1 ,as¡ 16 y 61 on the unit circle.

Theamplitudeof y= sinx is 1.

TRANSFORMATIONS OF THE SINE CURVE

In theDiscoveriesthat follow, we will consider different transformations of the sine curve y= sinx.We

will hence be able to generate the curve for the general sine function y=asinbx+c, a> 0 , b> 0.

Discovery 1 The family y=asinx, a> 0

Click on the icon to explore the family y=asinx, a> 0.

What to do:

1 Use the slider to vary the value ofa. Observe the changes to

the graph of the function.

2 Use the software to help complete the table:

a Function Maximum Minimum Period Amplitude

1 y= sinx 1 ¡ 1 2 ¼ 1

2 y= 2 sinx

3 y= 3 sinx

0 : 5 y=0:5 sinx

a y=asinx

3 How doesaaffect the function y=asinx?

Discovery 2 The family y= sinbx, b> 0

Click on the icon to explore the family y= sinbx, b> 0.

What to do:

1 Use the slider to vary the value ofb. Observe the changes to the graph of

the function.

DYNAMIC

SINE FUNCTION

DYNAMIC

SINE FUNCTION

xis measured in radians.

SINE FUNCTION

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yy

y=y=sinsinxx

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Y:\HAESE\CAM4037\CamAdd_09\231CamAdd_09.cdr Monday, 14 April 2014 5:59:20 PM BRIAN