## Periodic functions of the type y = a sin px

The graph of y = a sin(px) is similar to that of the sin graph of y = sin(x). Instead of varying or oscillating between +1 and -1, a sin(px) will vary between +a and -a. 'a' is called the amplitude and affects only the scale for y. The graph will repeat itself when the angle px increases by 360° or 2π radians. i.e. when x increases by 360°/p or 2π/p radians thus the period is 2π/p radians.

Thus 5 sin(2x) has a period of 2π/2 = π radians and amplitude 5. Hence there are 2 waves between 0° and 360°.

### Example

Sketch the curve for y = 3 sin(x+60°)

y = 3 sin(x+60°) is said to have a phase lead of 60° since its values are those of y = 3 sin(x) with x increased by 60°. y = 3 sin(x-60°) is said to lag by 60°.

## Solving Trig Equations

Here are some suggestions for solving trig identities:

- Change secant, cosecant and cotan into cos, sin and tan.
- rewrite in terms of sin and cos (use tan = sin/cos)
- If in second degree (ie cos
^{2}or sin^{2}) look for posible cos^{2}A + sin^{2}A = 1 relationship. - If it contains cos
^{2}or sin^{2}change sin^{2}into cos^{2}-1 and solve resulting quadratic. - If all this fails convert to an infinite series.
- If this fails plot as a graph.

## Plotting trig functions

### Example 1

y = 3 sin(2x) + sin x

### Example 2

y = 3 sin(x/2) - cos x