How Does the Unit Circle Work?

The unit circle is a circle with its center at $O = (0, 0)$ and a radius of 1. If you have a point $A$ on the unit circle and an angle $α$ spanning from the $x$ -axis to the line $O A$ , then the $x$ -coordinate of the point $A$ is $\cos α$ and the $y$ -coordinate of the point $A$ is $\sin α$ .

Theory

Exact Values for Sine, Cosine and Tangent


Degrees	Radians	$\sin α$	$\cos α$	$\tan α$

$0$ °	$0$	$0$	$1$	$0$

$30$ °	$\frac{π}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$

$45$ °	$\frac{π}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$

$60$ °	$\frac{π}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$

$90$ °	$\frac{π}{2}$	$1$	$0$	$not def.$

Here “not def.” is short for “not defined”. This is because these values give 0 in the denominator in the formula below, and division by zero is also not defined.

This table shows the most essential angles in radians and in degrees. It is very useful to learn which radians correspond to these angles. You are expected to know the exact values for the sine, cosine and tangent of each of these.

By the use of trigonometric identities we can expand the table to many more angles.

Formula

Tangent Expressed by Sine and Cosine

The value of $t a n$ can be found from this expression:

\tan x = \frac{\sin x}{\cos x}

For instance, you know that $\sin \frac{π}{4} = \frac{\sqrt{2}}{2}$ and that $\cos \frac{π}{4} = \frac{\sqrt{2}}{2}$ , so then:

\tan \frac{π}{4} = \frac{\sin \frac{π}{4}}{\cos \frac{π}{4}} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1

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How Do Radians Work?

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