Angles in all four quadrants

In a nutshell

The trigonometric ratios of $\sin$ , $\cos$ and $\tan$ can be given a geometric interpretation with the help of the unit circle centred at the origin. This lesson will cover this relationship, as well as the CAST method used to determine for which values of $\theta$ each of $\sin(\theta)$ , $\cos(\theta)$ and $\tan(\theta)$ take on positive or negative values.

Angles in the unit circle

Angles in the four quadrants can be represented by points on the unit circle, as shown.

$P$ is a point on the unit circle centred at $O$ .

$\theta$	The angle between the positive $x$ -axis and the line segment $OP$ , measured counterclockwise from the positive $x$ -axis.
$\sin(\theta)$	The $y$ -coordinate of $P$ .
$\cos(\theta)$	The $x$ -coordinate of $P$ .
$\tan(\theta)$	The slope of the line through the origin which passes through $P$ .

These descriptions give a procedure for finding the values of $\sin(\theta)$ , $\cos(\theta)$ or $\tan(\theta)$ for any value of $\theta$ .

Procedure

1.	Draw a line from the origin at an angle of $\theta$ degrees (measured anticlockwise from the positive $x$ -axis).
2.	Find the point where this line intersects the unit circle centred at the origin, call it $P$ .
3.	Identify the $x$ and $y$ values of $P$ - these are $\cos(\theta)$ and $\sin(\theta)$ respectively.
4.	Compute the ratio $\dfrac{\sin(\theta)}{\cos(\theta)}$ - this is $\tan(\theta)$ , which is also the gradient of the line.

Maths; Trigonometric equations; KS5 Year 12; Angles in all four quadrants

Negative values of $\theta$ are measured clockwise from the positive $x$ -axis. For example, when $\theta = -45^{\circ}$ , measure $45^{\circ}$ clockwise from the positive $x$ -axis to find $P = (\dfrac{\sqrt{2}}{2},-\dfrac{\sqrt{2}}{2})$ .

Note: For any angle $\theta$ , the point $P$ generated by the above procedure is the same as the point $P$ generated by $\theta \pm 360^{\circ}$ . This indicates that $\sin$ and $\cos$ are periodic, repeating every $360^{\circ}$ . This also allows you to convert any angle $\theta$ into an angle $\theta'$ such that $0^{\circ} \leq \theta' <360^{\circ}$ by repeatedly adding or subtracting multiplies of $360^{\circ}$ to $\theta$ .

Example 1

What is the value of $\tan(135^{\circ})$ ?

Draw a line at an angle of $135^{\circ}$  counterclockwise from the positive $x$ -axis, and observe that this line lies halfway between the negative $x$ -axis and the positive $y$ -axis. It must be the case that for any points $(x,y)$ lying on this line, $y = -x$ . It follows that the point $P = (-\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2})$ , and the slope of the line through the origin which passes through the point $P$ must be $-1$ .

Therefore, $\underline{\tan(135\degree) = -1}$ .

The CAST diagram

A simple way to determine whether $\sin(\theta)$ , $\cos(\theta)$ or $\tan(\theta)$ are positive or negative is via a handy mnemonic, where the plane is divided into four quadrants labelled with $\cos$ , all, $\sin$ and $\tan$ . This is referred to as the CAST diagram, as this is what the first letters of each quadrant spell when read counterclockwise starting from the bottom right.

Maths; Trigonometric equations; KS5 Year 12; Angles in all four quadrants

Each of $\sin(\theta)$ , $\cos(\theta)$ and $\tan(\theta)$ are positive only when $\theta$ is either in the quadrant labelled "All", or the quadrant they share a name with. For example, $\cos(\theta)$ is positive only in the bottom right and top right quadrants.

Example 2

Is $\sin(-60^{\circ})$ positive or negative?

$-60^{\circ} + 360^{\circ} = 300^{\circ}$ , so equivalently consider $\sin(300^{\circ})$ .

$270^{\circ} < 300^{\circ} < 360^{\circ}$ , so this falls into the $\cos$ quadrant of the CAST diagram.

Deduce that only $\cos(-60^{\circ})$ is positive, whereas $\sin(-60^{\circ})$ and $\tan(-60^{\circ})$ are negative.

Therefore, $\sin(-60\degree)$ is negative.

Acute angle identities

For any angle, there is an acute angle $0 \leq \theta \leq 90^{\circ}$ such that the angle is equivalent to one of $\theta$ , $180^{\circ} \pm \theta$ , or $360^{\circ} -\theta$ . You can use this with the CAST diagram to express $\sin$ , $\cos$ and $\tan$ in terms of $\sin(\theta)$ , $\cos(\theta)$ and $\tan(\theta)$ , where $\theta$ is the acute angle made with the $x$ -axis, via these rules:

$\begin{aligned}\sin(180^{\circ} \pm \theta) &= \mp \sin(\theta)\\\sin(360^{\circ}-\theta) &= -\sin(\theta)\\\\\cos(180^{\circ}\pm\theta) &= -\cos(\theta)\\\cos(360^{\circ}-\theta) &= \cos(\theta)\\\\\tan(180^{\circ}\pm\theta) &= \pm\tan(\theta)\\\tan(360^{\circ}-\theta) &= -\tan(\theta)\end{aligned}$

Maths; Trigonometric equations; KS5 Year 12; Angles in all four quadrants