New Zealand
Level 8 - NCEA Level 3

# Area between two sin/cos curves

Lesson

#### Example 1

When the graphs of the functions $\sin x$sinx and $\cos x$cosx are plotted, we see that there is a sequence of regions bounded by the two graphs. There is one such a bounded region between $x=0$x=0 and $x=2\pi$x=2π

To find its area, we need to know exactly where the region begins and ends. That is, we need to know where the two graphs intersect. We solve the equation $\sin x=\cos x$sinx=cosx.

In this example, we may recognise that the equation is satisfied when $x=\frac{\pi}{4}$x=π4 and when $x=\frac{5\pi}{4}$x=5π4. If not, we might manipulate the equation by dividing both sides by $\cos x$cosx to obtain $\tan x=1$tanx=1 and proceed from there. The diagram is shown below.

We carry out an integration to find the area of the region bounded by the sine and cosine curves between $\frac{\pi}{4}$π4 and $\frac{5\pi}{4}$5π4.

### Outcomes

#### M8-11

Choose and apply a variety of differentiation, integration, and antidifferentiation techniques to functions and relations, using both analytical and numerical methods

#### 91579

Apply integration methods in solving problems