Department of Mathematics


Worked Examples

The Unit Circle

We discussed trigonometric values of angles in a right-angle triangle, namely angles less than $90^{\circ}$ or $\pi/2$ rad. What about greater angles?

Consider the unit circle in the coordinate plane centered at the origin. Let $A$ be a point on the circle in the first quadrant and $\theta$ denote the angle it makes with the $x$-axis. Then the coordinates of $A$ are exactly $\left(x,y\right)=\left(\cos\left(\theta\right),\sin\left(\theta\right)\right)$. (Details)To see this, consider the perpendicular from $A$ to the $x$-axis, denote the endpoint $H$. The triangle $AHO$ is a right-angle triangle with side lengths of $OH$, $AH$, and $OA$ being $x$, $y$ and $1$ respectively. Then $\sin\left(\theta\right)=y/1=y$, $\cos\left(\theta\right)=x/1=x$.

We can extend this to any angle $\theta$: Think of the hands of a clock, one being fixed at the $x$-axis and the other going around counterclockwise in the unit circle to make an angle $\theta$ with the first one. The point the second hand reaches on the unit circle has coordinates $\left(\cos\left(\theta\right),\sin\left(\theta\right)\right)$.

Note: Going around the circle once measures an angle of $2 \pi$ rad. To get larger angles, we go around the circle more than once, say for $5 \pi/2$ we go around once and keep going until we reach the $y$-axis. To get negative angles, we go clockwise, say for $-\pi/2$, starting from the $x$-axis we go clockwise until reaching the $y$-axis.

Example. Find the trigonometric values of $\frac{\pi}{2}$.

At $\frac{\pi}{2}$, the hand of the clock is pointing at $\left(0,1\right)$. So, $\cos\left(\frac{\pi}{2}\right)=0, \sin\left(\frac{\pi}{2}\right)=1, \cot\left(\frac{\pi}{2}\right)=0/1=0$ and $\tan$ is not defined at $\frac{\pi}{2}$. A nice way of remembering which trigonometric functions take a positive or a negative value in each quadrant is the CAST rule:
C $\cos(\theta)$ is positive there
A all trigonometric functions are positive there
S $\sin(\theta)$ is positive there
T $\tan(\theta)$ is positive there

Some of the more important angles along with the corresponding coordinates are illustrated below.

Figure made by Gustavb at Wikipedia.

Example. Drag the angle slider on the bottom to see how trigonometric functions relate to the unit circle.