# Idea First we have [[trigonometric ratios]] that let us relate two sides of a triangle with an angle of that triangle. ![[20240404121026.png]] Second, we use the [[trigonometric ratios]] to determine the coordinate of a [[unit circle]] for every point on the circumference of that unit circle. For any coordinate pair on the circumference of the circle, the coordinate is given by $(cos(\theta), sin(\theta))$, where $\theta$ is the angle. ![[20240406202524.png]] Third, we can plot the input and output of the [[sine function]] and the cosine function, where $y = sin(\theta)$ is the **vertical** displacement or distance from the origin in the y-axis. Each coordinate pair is $(\theta, sin(\theta))$. $\theta$ is usually measured in [[radians]], and it is also referred to as time (hence $t$). Both functions are $2 \pi$ periodic. ![[SineAnim.gif]] Similarly, for the [[cosine function]], $y = cos(\theta)$ is the horizontal displacement of distance from the origin. It represents the x-coordinate on the unit circle. ![[CosineAnim.gif]] ![[Sine_and_cosine_animation.gif]] # References - [Sine and Cosine and Unit Circle](https://personal.morris.umn.edu/~mcquarrb/teachingarchive/Precalculus/Animations/SineCosineAnim.html)