Unit circle

In mathematics, a unit circle is a circle with a radius of 1. The equation of the unit circle is ${\displaystyle x^{2}+y^{2}=1}$. The unit circle is centered at the Origin, or coordinates (0,0). It is often used in Trigonometry.

The Unit Circle can be used to model every Trigonometric function.

Trigonometric functions in the unit circle

In a unit circle, where ${\displaystyle t}$ is the angle desired, ${\displaystyle x}$ and ${\displaystyle y}$ can be defined as ${\displaystyle \cos(t)=x}$ and ${\displaystyle \sin(t)=y}$. Using the function of the unit circle, ${\displaystyle x^{2}+y^{2}=1}$, another equation for the unit circle is found, ${\displaystyle \cos ^{2}(t)+\sin ^{2}(t)=1}$. When working with trigonometric functions, it is mainly useful to use angles with measures between 0 and ${\displaystyle \pi \over 2}$ radians, or 0 through 90 degrees. It is possible to have higher angles than that, however. Using the unit circle, two identities can be found: ${\displaystyle \cos(t)=\cos(2\cdot \pi k+t)}$ and ${\displaystyle sin(t)=\sin(2\cdot \pi k+t)}$ for any integer ${\displaystyle k}$.

The unit circle can substitute variables for trigonometric functions.