Trigonometry - AQATrigonometric graphs - Higher

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 \(y = \sin{\theta}\) has a maximum value of 1 and a minimum value of -1.

The graph of \(y = \cos{\theta}\) has a maximum value of 1 and a minimum value of -1.

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.

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

\(\sin{x} = 0.5\)

\(x = 30^\circ\)

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.

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

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

\(x\) = -330°, -210°, 30° and 150°.