When the sine of an angle is plotted against that angle measure,

the result is a classic "sine curve" shape.

the result is a classic "sine curve" shape.

Try this
Drag the
vertex of the triangle and see how the sine function varies with the angle.

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 above, 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.)

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.

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.

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.

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.

- Angle definition, properties of angles
- Standard position on an angle
- Initial side of an angle
- Terminal side of an angle
- Quadrantal angles
- Coterminal angles
- Reference angle

- Introduction to the six trig functions
- Functions of large and negative angles
- Inverse trig functions
- SOH CAH TOA memory aid
- Sine function (sin) in right triangles
- Inverse sine function (arcsin)
- Graphing the sine function
- Sine waves
- Cosine function (cos) in right triangles
- Inverse cosine function (arccos)
- Graphing the cosine function
- Tangent function (tan) in right triangles
- Inverse tangent function (arctan)
- Graphing the tangent function
- Cotangent function cot (in right triangles)
- Secant function sec (in right triangles)
- Cosecant function csc (in right triangles)

- The general approach
- Finding slant distance along a slope or ramp
- Finding the angle of a slope or ramp

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