How Does the Unit Circle Work?

The unit circle is a circle in 2 dimensions with center in $O=\left(0,0\right)$ and with radius 1. If you have a point $A$ on the unit circle and an angle $\alpha$ spanning from the $x$-axis to the line $OA$, then the $x$-coordinate of the point $A$ is $\mathrm{cos}\alpha$ and the $y$-coordinate of the point $A$ is $\mathrm{sin}\alpha$.

Theory

ExactValuesforSine,CosineandTangent

 Degrees Radians $\mathrm{sin}\alpha$ $\mathrm{cos}\alpha$ $\mathrm{tan}\alpha$ $0$° $0$ $0$ $1$ $0$ $30$° $\frac{\pi }{6}$ $\frac{1}{2}$ $\frac{\sqrt{3}}{2}$ $\frac{\sqrt{3}}{3}$ $45$° $\frac{\pi }{4}$ $\frac{\sqrt{2}}{2}$ $\frac{\sqrt{2}}{2}$ $1$ $60$° $\frac{\pi }{3}$ $\frac{\sqrt{3}}{2}$ $\frac{1}{2}$ $\sqrt{3}$ $90$° $\frac{\pi }{2}$ $1$ $0$

Here “not def.” is short for “not defined”. This is because these values give 0 in the denominator in the formula below.

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

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

Formula

TangentExpressedbySineandCosine

The value of tan can be found from this expression:

 $\mathrm{tan}x=\frac{\mathrm{sin}x}{\mathrm{cos}x}.$

For instance, you know that $\mathrm{sin}\frac{\pi }{4}=\frac{\sqrt{2}}{2}$ and that $\mathrm{cos}\frac{\pi }{4}=\frac{\sqrt{2}}{2}$, then tan is

 $\mathrm{tan}\frac{\pi }{4}=\genfrac{}{}{1.0pt}{}{\phantom{\rule{0.17em}{0ex}}\mathrm{sin}\frac{\pi }{4}\phantom{\rule{0.17em}{0ex}}}{\phantom{\rule{0.17em}{0ex}}\mathrm{cos}\frac{\pi }{4}\phantom{\rule{0.17em}{0ex}}}=\genfrac{}{}{1.0pt}{}{\phantom{\rule{0.17em}{0ex}}\frac{\sqrt{2}}{2}\phantom{\rule{0.17em}{0ex}}}{\phantom{\rule{0.17em}{0ex}}\frac{\sqrt{2}}{2}\phantom{\rule{0.17em}{0ex}}}=1.$