# Graph of the Sine function

When the sine of an angle is plotted against that angle measure,
the result is a classic “sine curve” shape.

To graph the sine function, we mark the angle along the horizontal x axis, and for each angle, we put the sine of that angle on the vertical y-axis. The result, as seen above, is a smooth curve that varies from +1 to -1.

Curves that follow this shape are called ‘sinusoidal’ after the name of the sine function. This shape is also called a sine wave, especially when it appears in radio and electronic circuits.

In the diagram at the link, drag the point A around in a circular path to vary the angle CAB. As you do so, the point on the graph moves to correspond with the angle and its sine. (If you check the “progressive mode” box, the curve will be drawn as you move the point A instead of tracing the existing curve.)

## The domain of the sine function

As you drag the point A around notice that after a full rotation about B, the graph shape repeats. The shape of the sine curve is the same for each full rotation of the angle and so the function is called ‘periodic’. The period of the function is 360° or 2π radians. You can rotate the point as many times as you like. This means you can find the sine of any angle, no matter how large. In mathematical terms we say the ‘domain’ of the sine function is the set of all real numbers.

## Range

The range of a function is the set of result values it can produce. The sine function has a range that goes from -1 to +1. Looking at the sine curve you can see it never goes outside this range.

## The inverse sine function What if we were asked to find the inverse sine of a number, let’s say 0.5? In other words, what angle has a sine of 0.5?

If we look at the curve above we see four angles whose sine is 0.5 (red dots). In fact, since the graph goes on forever in both directions, there are an infinite number of angles that have a sine of a 0.5.

### So what does a calculator say?

If you ask a calculator to find the arcsine (sin-1) of a number, it cannot return an infinitely long list of angles, so by convention it finds just the first one. But remember, there are many others.