Trigonometric graphs - Higher

Circle with triangle from centre to edge, at angle theta

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 x-coordinate is \cos{\theta} and the y-coordinate is \sin{\theta}.

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

The values of \cos{\theta} and \sin{\theta} also change.

The graphs of y = \sin{\theta} and y = \cos{\theta} can be plotted.

The graph of y = sin θ

Graph of y = sin theta

The graph of y = \sin{\theta} 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

Graph of y = cos theta

The graph of y = \cos{\theta} 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 \cos{\theta} and \sin{\theta} change, therefore the value of \tan{\theta} will change.

Y = tan theta

This graph has a period of 180°.

Calculating angles from trigonometric graphs

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

Example

Solve the equation \sin{x} = 0.5 for all values of x between -360^\circ \leq x \leq 360^\circ.

\sin{x} = 0.5

Using a calculator gives one solution:

x = 30^\circ

Draw the horizontal line y = 0.5.

The line y = 0.5 crosses the graph of y = \sin{x} four times in the interval -360^\circ \leq \theta \leq 360^\circ so there are four solutions.

Graph of y=sin x

There is a line of symmetry at x = 90^\circ, so there is a solution at 180 - 30 = 150^\circ.

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

The solutions to the equation \sin{x} = 0.5 are:

x = -330°, -210°, 30° and 150°.

Question

Solve the equation \cos{x} = 0.75 for all values of x between -360^\circ \leq x \leq 360^\circ. Give your answer to the nearest degree.

\cos{x} = 0.75

Using a calculator gives one solution:

x = 41^\circ (to the nearest degree)

Draw the horizontal line y = 0.75.

The line y = 0.75 crosses the graph of y = \cos{x} four times in the interval -360^\circ \leq x \leq 360^\circ so there are four solutions.

Graph of y=cos x

There is a line of symmetry at x = 0^\circ, 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 \cos{x} = 0.75 are:

x = -319°, -41°, 41° and 319°.

Question

Given that \tan{60} = \sqrt{3}, calculate the other values of x in the interval 0^\circ \leq x \leq 720^\circ for which \tan{x} = \sqrt{3}.

Square root of three is shown as 1.7 on the y axis

The period of the graph y = \tan{x} is 180° so to calculate the other solutions add 180°.

x = 60°, 240°, 420° and 600°.