Cambridge International AS and A Level Mathematics Pure Mathematics 1

Trigonometry

P1^

7

10  Prove the identity^1

1

+ + 2

+

sin ≡

cos

cos

sincos

x

x

x

xx

[Cambridge AS & A Level Mathematics 9709, Paper 1 Q2 November 2008]

11  Prove the identity sin

sin

sin

sin

x tan.

x

x

x

x

11

2 2

−

−

+

≡

[Cambridge AS & A Level Mathematics 9709, Paper 1 Q1 June 2009]

The sine and cosine graphs

In figure 7.12, angles have been drawn at intervals of 30° in the unit circle, and

the resulting y co-ordinates plotted relative to the axes on the right. They have

been joined with a continuous curve to give the graph of sin θ for 0°  θ  360°.

The angle 390° gives the same point P 1 on the circle as the angle 30°, the angle

420° gives point P 2 and so on. You can see that for angles from 360° to 720° the

sine wave will simply repeat itself, as shown in figure 7.13. This is true also for

angles from 720° to 1080° and so on.

Since the curve repeats itself every 360° the sine function is described as periodic,

with period 360°.

In a similar way you can transfer the x co-ordinates on to a set of axes to obtain

the graph of cos θ. This is most easily illustrated if you first rotate the circle

through 90° anticlockwise.

O

P 3 P 3

P 9

90° 270°

y

x 180° 360°

+1

–1

P 4 P 2

P 8 P 10

P 1

P 1

P 5

P 7 P 11

P 0

P 12 P 0

P 9

P 10

P 8 P 11

P 12

P 6

P 4

P 5

P 7

P 6

P 2

sin θ

θ

Figure 7.12

O

sin θ

180° 360° 540° 720° θ

+1

–1

Figure 7.13