Unit Circle Unit Circle

Introduction

The unit circle is a circle with a radius of 1, centered at the origin (0, 0) in the Cartesian coordinate system.

x(\theta) = \cos(\theta) \\ y(\theta) = \sin(\theta)
We use the unit circle to define the trigonometric functions sine and cosine.

For any angle
\theta
:
The
x
-value of the point on the unit circle is
\cos(\theta)
.
The
y
-value of the point on the unit circle is
\sin(\theta)
.

If the point is on the right half, the
x
-value is positive.
\cos(\theta)
is positive.
If the point is on the left half, the
x
-value is negative.
\cos(\theta)
is negative.
If the point is on the top half, the
y
-value is positive.
\sin(\theta)
is positive.
If the point is on the bottom half, the
y
-value is negative.
\sin(\theta)
is negative.

Radians vs Degrees

Angles can be measured in degrees or radians. We use radians:

360^\circ = 2\pi \text{ rad}
\text{radians} = \text{degrees} \cdot \frac{\pi}{180^\circ}

Key Angles

Here are some key angles and their corresponding coordinates on the unit circle in all four quadrants:

\begin{align*} & \sin(0) & = &&& 0 &\quad\quad&& \cos(0) && = &&& 1 \\\\ & \sin\left(\frac{\pi}{6}\right) & = &&& \frac{1}{2} &\quad\quad&& \cos\left(\frac{\pi}{6}\right) && = &&& \frac{\sqrt{3}}{2} \\\\ & \sin\left(\frac{\pi}{4}\right) & = &&& \frac{\sqrt{2}}{2} &\quad\quad&& \cos\left(\frac{\pi}{4}\right) && = &&& \frac{\sqrt{2}}{2} \\\\ & \sin\left(\frac{\pi}{3}\right) & = &&& \frac{\sqrt{3}}{2} &\quad\quad&& \cos\left(\frac{\pi}{3}\right) && = &&& \frac{1}{2} \\\\ & \sin\left(\frac{\pi}{2}\right) & = &&& 1 &\quad\quad&& \cos\left(\frac{\pi}{2}\right) && = &&& 0 \\\\ & \sin\left(\frac{2\pi}{3}\right) & = &&& \frac{\sqrt{3}}{2} &\quad\quad&& \cos\left(\frac{2\pi}{3}\right) && = &&& -\frac{1}{2} \\\\ & \sin\left(\frac{3\pi}{4}\right) & = &&& \frac{\sqrt{2}}{2} &\quad\quad&& \cos\left(\frac{3\pi}{4}\right) && = &&& -\frac{\sqrt{2}}{2} \\\\ & \sin\left(\frac{5\pi}{6}\right) & = &&& \frac{1}{2} &\quad\quad&& \cos\left(\frac{5\pi}{6}\right) && = &&& -\frac{\sqrt{3}}{2} \\\\ & \sin\left(\pi\right) & = &&& 0 &\quad\quad&& \cos\left(\pi\right) && = &&& -1 \\\\ & \sin\left(\frac{7\pi}{6}\right) & = &&& -\frac{1}{2} &\quad\quad&& \cos\left(\frac{7\pi}{6}\right) && = &&& -\frac{\sqrt{3}}{2} \\\\ & \sin\left(\frac{5\pi}{4}\right) & = &&& -\frac{\sqrt{2}}{2} &\quad\quad&& \cos\left(\frac{5\pi}{4}\right) && = &&& -\frac{\sqrt{2}}{2} \\\\ & \sin\left(\frac{4\pi}{3}\right) & = &&& -\frac{\sqrt{3}}{2} &\quad\quad&& \cos\left(\frac{4\pi}{3}\right) && = &&& -\frac{1}{2} \\\\ & \sin\left(\frac{3\pi}{2}\right) & = &&& -1 &\quad\quad&& \cos\left(\frac{3\pi}{2}\right) && = &&& 0 \\\\ & \sin\left(\frac{5\pi}{3}\right) & = &&& -\frac{\sqrt{3}}{2} &\quad\quad&& \cos\left(\frac{5\pi}{3}\right) && = &&& \frac{1}{2} \\\\ & \sin\left(\frac{7\pi}{4}\right) & = &&& -\frac{\sqrt{2}}{2} &\quad\quad&& \cos\left(\frac{7\pi}{4}\right) && = &&& \frac{\sqrt{2}}{2} \\\\ & \sin\left(\frac{11\pi}{6}\right) & = &&& -\frac{1}{2} &\quad\quad&& \cos\left(\frac{11\pi}{6}\right) && = &&& \frac{\sqrt{3}}{2} \\\\ & \sin\left(2\pi\right) & = &&& 0 &\quad\quad&& \cos\left(2\pi\right) && = &&& 1 \\\\ \end{align*}