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 \(\sin{\theta}\) where \({\theta}\) is measured anti-clockwise from the positive \(x\)-axis.

As the point P moves anticlockwise round the circle from (1, 0), the angle \(\theta\) increases until P returns to its starting position at (1, 0) when \(\theta\) = 360°. If P continues moving past (1, 0), \(\theta\) becomes greater than 360°, and the next time P is at (1, 0), \(\theta\) will be 720°. And so on. Instead of P moving anticlockwise from (1, 0), if it goes clockwise then \(\theta\) will be negative!

No matter where P is on the circle, the \(x\)-coordinate gives the value of \(\cos{\theta}\) and the \(y\)-coordinate gives the value of \(\sin{\theta}\). Thus, the values of \(\cos{\theta}\) and \(\sin{\theta}\) will sometimes be positive and sometimes negative depending on the value of \(\theta\).

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.

This is defined as \(\tan{\theta} = \frac{o}{a}\) and from the circle \(o = \sin{\theta}\) and \(a = \cos{\theta}\).

\(\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}}\)

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°.