# Sine and Cosine intuition One of the challenges with understanding sine and cosine is that they are *defined* to be functions of $\theta$, yeah they are computed via lengths of a triangle that $\theta$ is a part of. For instance look at the image below:  We see that our sine and cosine are defined as: $sin(\theta) = \frac{b}{c}$ $cos(\theta) = \frac{a}{c}$ And that right there is what subtly produces confuses! Generally, if we are working with a function $f$ and we state that it's input domain is $x$, $x$ tends to show up in the output. For example: $f(x) = x^2 + 4$ $f(x) = e^x$ $f(x) = x^8 - x^6 + 3x$ $f(x) = 5sin(x)$ In all of the above examples, it is *explicitly* written down that $x$ is indeed involved in the computation of $f(x)$ for a given $x$. In the example of $sin$ and $cos$, that is not the case! We state that $sin(\theta)$ is a function of $\theta$, and then when we look at the actual computation that is to be performed, no $\theta$ is present! The only thing we see is a fraction involving some $b$ and $c$. This is because $sin$ and $cos$ are defined with respect to a specific structure, either a right angle triangle or a unit circle (they can be treated as equivalent). So, in a *sense*, it would be appropriate to define as: $sin(a,b,c, \theta) = \frac{b}{c}$ $cos(a,b,c, \theta) = \frac{a}{c}$ This definition make it explicit what $sin$ and $cos$ are calculated in reference to. However, the reason that this is not necessary is because, *baked into our traditional definition* of $sin$ and $cos$ are the following: * Both are defined with respect to a right triangle (it can either be a stand alone right triangle, or one that is based on the $x$ and $y$ components of a point on a unit circle) * Due to the *constraint* above (that we are dealing with a right triangle), it is often just taken as common knowledge that our angle $\theta$ is what will cause our triangle side lengths to vary, so in a very real sense $\theta$ can be thought of as the main input. The key idea to keep in mind here is that $sin$ and $cos$ are functions that most definitely encode a decent amount of background information! You must keep in mind that they were defined in terms of a right angle triangle, to calculate a specific value based on a specific $\theta$ you must know the side lengths, as defined via a walk around the unit circle. ## We may not want to interpret sin as part of circle * See [here](https://betterexplained.com/articles/intuitive-understanding-of-sine-waves/). * Also, sin is 1 dimensional! (this was always clear) * Circles are *examples* of sines, *not* their origin/source. * Sine is a cycle, a repeating pattern # Origin of periodic interpretation Below, note that $sin$ is the vertical component of our green line, and $cos$ is the horizontal component.  --- Tags: #trigonometry References: * Euler's Formula - 03172021, Notability * [Intuitive understanding of sin waves](https://betterexplained.com/articles/intuitive-understanding-of-sine-waves/)