# Trigonometric graphs - Higher

This circle has the centre at the origin and a radius of 1 unit.

The point P can move around the circumference of the circle. At point P the -coordinate is and the -coordinate is .

As the point P moves anticlockwise around the circle from 0°, the angle changes.

The values of and also change.

The graphs of and can be plotted.

## The graph of y = sin θ

The graph of has a maximum value of 1 and a minimum value of -1.

The graph has a period of 360°. This means that it repeats itself every 360°.

## The graph of y = cos θ

The graph of has a maximum value of 1 and a minimum value of -1.

The graph has a period of 360°.

## The graph of y = tan θ

As the point P moves anticlockwise round the circle, the values of and change, therefore the value of will change.

This graph has a period of 180°.

## Calculating angles from trigonometric graphs

The symmetrical and periodic properties of the trigonometric graphs will give an number of solutions to any trigonometric equation.

### Example

Solve the equation for all values of between . Using a calculator gives one solution: Draw the horizontal line .

The line crosses the graph of four times in the interval so there are four solutions.

There is a line of symmetry at , so there is a solution at .

The period is 360° so to find the next solutions subtract 360°.

The solutions to the equation are: = -330°, -210°, 30° and 150°.

Question

Solve the equation for all values of between . Give your answer to the nearest degree. Using a calculator gives one solution: (to the nearest degree)

Draw the horizontal line .

The line crosses the graph of four times in the interval so there are four solutions.

There is a line of symmetry at , so there is a solution at -41°.

The period is 360° so to find the other solutions add and subtract 360°.

The solutions to the equation are: = -319°, -41°, 41° and 319°.

Question

Given that , calculate the other values of in the interval for which .

The period of the graph is 180° so to calculate the other solutions add 180°. = 60°, 240°, 420° and 600°.