# Trigonometry

## Preface

Trigonometry is not about triangles, it’s all about *circles*.

## Slope

Represents how a line rises or falls.

## Radian

A radian starts from 180 degree (far left `x`

, aka `-x`

) in the second quadrant and goes clockwise.

$\pi$ is the ratio of a circle’s circumference to its diameter, approximately `3.14159`

.

There are $2\pi$ radians in a full circle, where

$\dfrac{\pi}{3} \text{ radians} = \text{ 60 degrees }$ $\pi \text{ radians} = \text{ 180 degrees }$ $2\pi \text{ radians} = \text{ 360 degrees }$It represents the ratio of the circumference of a circle to its radius.

Since we have the ratio we can use the radius to find the actual value of the circumference w/

$C = 2\pi r$And we can find the area of it by

$\begin{split} \text{Area } &= \text{length } \cdot \text{ breadth} \\ &= \pi r \cdot r \\ &= \pi r^2 \end{split}$## Sin & Cos

`sin(0)`

is measuring the distance of `y`

and `cos(0)`

is measuring the distance of the `x`

coordinate as you walk around the circle in an anti-clockwise fashion.

both will inevitably have the exact same graph because they cycle through the quadrants in the same manner, just delayed. However, since they start at different points, `y`

and `x`

, it means the graphs are translated. So if we started from the top of the circle at our `y`

then `x = 0`

whereas if we started at far right for `x`

then `y = 0`

. So there is a translation of `1`

.

When they go from `1`

to `-1`

what is really happening is the trough of the graph is peaking but inverted since turning it into a negative flips the graph.

If we think of the question: what would the `sin`

and `cos`

be if our radian was `3`

in a circle?

Well we know `x = cos`

and `y = sin`

so as we start from the far

## Tan

`tan`

is a shorterned name from tangent because the point of it is where the slope is tangent to the point on the circumference point of the circle.

`tan(0)`

is measuring the distance of the radius, that is made up from `sin(0)`

and `cos(0)`

. And since the circle’s points are `x = 1`

and `-x = -1`

the set of all solution points is the graph of the equation.