Cambridge International AS and A Level Mathematics Pure Mathematics 1

P1^

7

The sine and cosine graphs

Figure 7.14 shows the circle in this new orientation, together with the resulting

graph.

For angles in the interval 360°  θ  720°, the cosine curve will repeat itself. You

can see that the cosine function is also periodic with a period of 360°.

Notice that the graphs of sin θ and cos θ have exactly the same shape. The cosine

graph can be obtained by translating the sine graph 90° to the left, as shown in

figure 7.15.

From the graphs it can be seen that, for example

cos 20° = sin 110°, cos 90° = sin 180°, cos 120° = sin 210°, etc.

In general

cos θ ≡ sin (θ  + 90°).

●?^1 What do the graphs of sin θ and cos θ look like for negative angles?

2 Draw the curve of sin θ for 0°  θ  90°.

Using only reflections, rotations and translations of this curve, how can you

generate the curves of sin θ and cos θ for 0°  θ  360°?

P O

(^3) P 3
P^9
cos θ
y θ
x
180° 360°
+1
–1
P^2
P^4
P^10
P^8
P^1
P 1
P^5
P^11
P^7
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
90° 270°
Figure 7.14 
–1
O 20° 90° θ
110° 210°
120° 270°
y = cos θ
y = sin θ
180° 360°
+1
y
Figure 7.15