## Section13.1Sine and Cosine Functions

You have seen sines and cosines in the context of triangles. There, we define them using the sides of a right-angled triangle.

\begin{align*} \amp \sin\,\theta = \dfrac{p}{h}, \\ \amp\cos\,\theta = \dfrac{b}{h}. \end{align*}
In a right-angled triangle, the angle can go up to $90^{\circ}\text{.}$ More generally, the angle can be extended to the full $360^{\circ}$ in the plane, which we have used to indicate direction in a plane.

When we think sines and cosines as functions of $\theta\text{,}$ we usually view $\theta$ in radian and not in degrees with $360^{\circ}$ equal to $2\pi$ radian. Furthermore, we extend the range of values for $\theta$ to any real value, $-\infty \lt \theta \lt \infty \text{,}$ and require that sines and cosines be periodic in $\theta$ with period $2 \pi\text{.}$

\begin{align*} \amp \sin\,(\theta + 2\pi m) = \sin\,\theta, \ \ \ (m\text{ any integer}) \\ \amp\cos\,(\theta + 2\pi n) = \cos\,\theta, \ \ \ (n\text{ any integer}) \end{align*}

This periodicity of sine and cosine functions makes them valuable for representing vibrations and waves, such as sound and light.

The uniform circular motion in the $xy$ plane has two harmonic motions inside it - one along the $x$ axis and one along the $y$ axis. Here is website that you can play aroud interactively Sine and Cosine in Uniform Motion in a Unit Circle or go to "https://www.desmos.com/calculator/cpb0oammx7".