# Trigonometry ### Link between unit circles and triangles Trigonometry is defined as follows: > The branch of mathematics that studies relationships between side lengths and angles of triangles. In this light we are frequently dealing with sines, cosines, and tangents. But here is the interesting thing: sines, cosines and tangents are also highly coupled to **circular motion**. Now that at first may seem odd: triangles and circles seem in some ways to as opposite as can be. On one hand a triangle is the 2-dimensional polygon with the *minimum* possible number of sides. On the other hand, a circle is the object that a polygon approaches as the number of sides become infinite. We can view these as a lower bound and upper bound on the space of polygons in 2-dimensions. So it may at first seem slightly surprising that they are indeed connected! Where does this connection arise from. Well it is only fitting that it a geometric visualization makes this more clear. Below we have a unit *circle* with a *triangle* inside it:  Ahah! And just like that it is clear: the trig functions are specifically defined with respect to a unit circle. By letting the hypotenuse move freely (always based at the origin but able to move around the circle remaining at a distance of 1 from origin) the angle, $\theta$, changes. We see that any given $\theta$ can be mapped to different ratios of the sides of this embedded triangle. For instance, we *define* the trig functions to be: $sin(\theta) = \frac{\text{opposite side length}}{\text{hypotenuse side length}}$ $cos(\theta) = \frac{\text{adjacent side length}}{\text{hypotenuse side length}}$ And so on. The key idea here is a certain $\theta$ provides a *constraint*; once you know $\theta$ is 30 for instance, it has constrained the triangle side lengths. We simply state that the sine function is the ratio of the opposite over the hypotenuse side length for a given theta. We can see all of the trig functions below:   Now, we can see how this is so linked to circular motion! Any time we have some type of *circular* or *cyclic* motion (a pendulum, a spring moving up and down, the temperature over the course of the year, and so on) trigonometry provides us with a great set of mathematical tools to *describe* this phenomena. --- Date: 20210814 Links to: [Mathematics MOC](Mathematics%20MOC.md) Tags: #review References: * []()