Unit Circle

What is the unit circle used for?

For any point on the circumference of the unit circle, the x-coordinate will be its cos value, and the y-coordinate will be the sin value. Therefore, the unit circle can help us find the values of the trigonometric functions sin, cos and tan for certain points. We can draw the unit circle for commonly used Angles to find out their sin and cos values.

The unit circle Image: public domain

The unit circle has four quadrants: the four regions (top right, top left, bottom right, bottom left) in the circle. As you can see, each quadrant has the same sin and cos values, only with the signs changed.

How to derive sine and cosine from the unit circle

Let's look at how this is derived. We know that when 𝜃 = 0 ° , sin𝜃 = 0 and cos𝜃 = 1. In our unit circle, an angle of 0 would look like a straight horizontal line:

The unit circle for 𝜃 = 0

Therefore, as sin𝜃 = 0 and cos𝜃 = 1, the x-axis has to correspond to cos𝜃 and the y-axis to sin𝜃. We can verify this for another value. Let's look at 𝜃 = 90 ° or 𝜋 / 2.

The unit circle for 𝜃 = 90

In this case, we have a straight vertical line in the circle. We know that for 𝜃 = 90 ° , sin 𝜃 = 1 and cos 𝜃 = 0. This corresponds to what we found earlier: sin 𝜃 is on the y-axis, and cos 𝜃 is on the x-axis. We can also find tan 𝜃 on the unit circle. The value of tan 𝜃 corresponds to the length of the line that goes from the point on the circumference to the x-axis. Also, remember that tan𝜃 = sin𝜃 / cos𝜃.

The unit circle and Pythagorean identity

From Pythagoras' theorem, we know that for a right-angled triangle $a^2+b^2=c^2$. If we were to construct a right angled triangle in a unit circle, it would look like this:

The unit circle with sin and cos

So a and b are sin𝜃, and cos𝜃 and c is 1. Therefore we can say: ${\sin }^2𝜃+{\cos }^2𝜃=1$ which is the first Pythagorean identity.