notes/education/math/MATH1060 (trig)/Graphing.md

# Sine/Cosine
![A graph of sine and cosine](./assets/graphsincos.png)

Given the above graph:
- At the origin, $sin(x) = 0$ and $cos(x) = 1$
- A full wavelength takes $2\pi$

# Manipulation
| Formula          | Movement                           |
| ---------------- | ---------------------------------- |
| $y = cos(x) - 1$ | Vertical shift down by 1           |
| $y = 2cos(x)$    | Vertical stretch by a factor of 2  |
| $y = -cos(x)$    | Flip over x axis                   |
| $y = cos(2x)$    | Horizontal shrink by a factor of 2 |
# Periodic Functions
A function is considered periodic if it repeats itself at even intervals, where each interval is a complete cycle, referred to as a *period*.
# Sinusoidal Functions
A function that has the same shape as a sine or cosine wave is known as a sinusoidal function.

There are 4 general functions:

| $$A * sin(B*x - C) + D$$                  | $$ y = A * cos(B*x -c) + D$$           |
| ----------------------------------------- | -------------------------------------- |
| $$ y = A * sin(B(x - \frac{C}{B})) + D $$ | $$ y = A*cos(B(x - \frac{C}{B})) + D$$ |


How to find the:
- Amplitude: $|A|$
- Period: $\frac{2\pi}{B}$
- Phase shift: $\frac{C}{|B|}$
- Vertical shift: $D$ 


$$ y = A * \sin(B(x-\frac{C}{B})) $$
# Tangent/Cotangent
$$ y = tan(x) $$
![Graph of tangent](assets/graphtan.png)
To find relative points to create the above graph, you can use the unit circle:

If $tan(x) = \frac{sin(x)}{cos(x})$, then:

| $sin(0) = 0$                              | $cos(0) = 1$                              | $tan(0) = \frac{cos(0)}{sin(0)} = \frac{0}{1} =0$                 |
| ----------------------------------------- | ----------------------------------------- | ----------------------------------------------------------------- |
| $sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$ | $cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$ | $tan(\frac{\pi}{4} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}{}}$ |
| $sin(\frac{\pi}{2}) = 1$                  | $cos(\frac{\pi}{2}) = 0$                  |                                                                   |

$$ y = cot(x) $$
![Graph of cotangent](assets/graphcot.svg)