Trigonometric Functions and the Unit Circle

We have already defined the trigonometric functions in terms of right triangles. In this section, we will redefine them in terms of the unit circle. Recall that a unit circle is a circle centered at the origin with radius 1. The angle $t$ (in radians) forms an arc of length $s$.

The x- and y-axes divide the coordinate plane (and the unit circle, since it is centered at the origin) into four quarters called quadrants. We label these quadrants to mimic the direction a positive angle would sweep. The four quadrants are labeled I, II, III, and IV.

For any angle $t$, we can label the intersection of its side and the unit circle by its coordinates, $(x, y)$. The coordinates $x$ and $y$ will be the outputs of the trigonometric functions $f(t) = \cos t$ and $f(t) = \sin t$, respectively. This means:

$\displaystyle{ \begin{aligned} x &= \cos t \\ ​y &= \sin t \end{aligned} }$ 

The diagram of the unit circle illustrates these coordinates.  

Unit circle

Coordinates of a point on a unit circle where the central angle is $t$ radians.

Note that the values of $x$ and $y$ are given by the lengths of the two triangle legs that are colored red. This is a right triangle, and you can see how the lengths of these two sides (and the values of $x$ and $y$) are given by trigonometric functions of $t$.

For an example of how this applies, consider the diagram showing the point with coordinates $\displaystyle{ \left(-\frac{\sqrt2}{2}, \frac{\sqrt2}{2}\right) }$ on a unit circle.

Point on a unit circle

The point $\displaystyle{ \left(-\frac{\sqrt2}{2}, \frac{\sqrt2}{2}\right) }$ on a unit circle.

We know that, for any point on a unit circle, the $x$-coordinate is $\cos t$ and the $y$-coordinate is $\sin t$. Applying this, we can identify that $\displaystyle{ \cos t = -\frac{\sqrt2}{2} }$ and  $\displaystyle{\sin t = -\frac{\sqrt2}{2}}$ for the angle $t$ in the diagram. 

Recall that $\displaystyle{\tan t = \frac{\sin t}{\cos t}}$.  Applying this formula, we can find the tangent of any angle identified by a unit circle as well. For the angle $t$ identified in the diagram  of the unit circle showing the point $\displaystyle{ \left(-\frac{\sqrt2}{2}, \frac{\sqrt2}{2}\right) }$, the tangent is:

$\displaystyle{ \begin{aligned} \tan t &= \frac{\sin t}{\cos t} \\ &= \frac{-\frac{\sqrt2}{2}}{-\frac{\sqrt2}{2}} \\ &= 1 \end{aligned} }$

We have previously discussed trigonometric functions as they apply to right triangles. This allowed us to make observations about the angles and sides of right triangles, but these observations were limited to angles with measures less than $90^{\circ}$. Using the unit circle, we are able to apply trigonometric functions to angles greater than $90^{\circ}$.

Further Consideration of the Unit Circle

The coordinates of certain points on the unit circle and the the measure of each angle in radians and degrees are shown in the unit circle coordinates diagram. This diagram allows one to make observations about each of these angles using trigonometric functions.

Unit circle coordinates

The unit circle, showing coordinates and angle measures of certain points.

We can find the coordinates of any point on the unit circle. Given any angle $t$, we can find the $x$- or $y$-coordinate at that point using $x = \text{cos } t$  and $y = \text{sin } t$.

The unit circle demonstrates the periodicity of trigonometric functions. Periodicity refers to the way trigonometric functions result in a repeated set of values at regular intervals. Take a look at the $x$-values of the coordinates in the unit circle above for values of $t$ from $0$ to $2{\pi}$:

${1, \frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2}, \frac{1}{2}, 0, -\frac{1}{2}, -\frac{\sqrt{2}}{2}, -\frac{\sqrt{3}}{2}, -1, -\frac{\sqrt{3}}{2}, -\frac{\sqrt{2}}{2}, -\frac{1}{2}, 0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1}$

We can identify a pattern in these numbers, which fluctuate between $-1$ and $1$. Note that this pattern will repeat for higher values of $t$. Recall that these $x$-values correspond to $\cos t$. This is an indication of the periodicity of the cosine function.

Example 

Solve $\displaystyle{ \sin{ \left(\frac{7\pi}{6}\right) } }$. 

It seems like this would be complicated to work out. However, notice that the unit circle diagram shows the coordinates at $\displaystyle{ t = \frac{7\pi}{6} }$. Since the $y$-coordinate corresponds to $\sin t$, we can identify that 

$\displaystyle{\sin{ \left(\frac{7\pi}{6}\right)} = -\frac{1}{2} }$