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
$$ y = tan(x) $$
![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:

| $sin(0) = 0$                              | $cos(0) = 1$                              | $tan(0) = \frac{cos(0)}{sin(0)} = \frac{0}{1} =0$                |
| ----------------------------------------- | ----------------------------------------- | ---------------------------------------------------------------- |
| $sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$ | $cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$ | $tan(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}/\frac{\sqrt{2}}{2} = 1$ |
| $sin(\frac{\pi}{2}) = 1$                  | $cos(\frac{\pi}{2}) = 0$                  | $tan(\frac{\pi}{2}) = \frac{1}{0} = DNF$                         |
Interpreting the above table:
- When $x = 0$, $y = 0$
- When $x = \frac{\pi}{4}$, $y = 1$
- When $x = \frac{\pi}{2}$, there's an asymptote

Without any transformations applied, the period of $tan(x) = \pi$. Because $tan$ is an odd function, $tan(-x) = -tan(x)$. 
# Cotangent
$$ y = cot(x) $$
![Graph of cotangent](assets/graphcot.svg)

To find relative points to create the above graph, you can use the unit circle:

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

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

Without any transformations applied, the period of $cot(x) = \pi$. Because $cot$ is an odd function, $cot(-x) = -cot(x)$. 

# Features of Tangent and Cotangent
Given the form $y = A\tan(Bx - C) + D$ (the same applies for $\cot$)
- The stretching factor is $|A|$
- The period is $\frac{\pi}{|B|}$
- The domain of $tan$ is all of $x$, where $x \ne \frac{C}{B} + \frac{\pi}{2} + {\pi}{|B|}k$, where $k$ is an integer. (everywhere but the asymptotes)
- The domain of $cot$ is all of $x$, where $x \ne \frac{C}{B} + \frac{\pi}{|B|}k$, where $k$ is an integer (everywhere but the asymptotes)
- The range of both is $(-\infty, \infty)$
- The phase shift is $\frac{C}{B}$
- The vertical shift is $D$

# Secant
$$ y = \sec(x) $$
![Graph of secant](assets/graphsec.jpg)

$$ sec(x) = \frac{1}{\cos{x}} $$
Because secant is the reciprocal of cosine, when $\cos{x} = 0$, then secant is undefined. $|\cos$| is never *greater than* 1, so secant is never *less than* 1 in absolute value. When the graph of cosine crosses the x axis, an asymptote for a matching graph of secant will appear there. 

The general form of secant is:
$$ y = A\sec(B{x} - C) + D $$
$A$, $B$, $C$, and $D$ will have similar meanings to the secant function as they did to the sine and cosine functions. 

# Cosecant
$$ y = \csc(x) $$
![Graph of cosecant](assets/graphsec.jpg)

$$ \csc(x) = \frac{1}{\sin(x)} $$

Because cosecant is the reciprocal of sine, when $\sin{x} = 0$, then cosecant is undefined. $|\sin$| is never *greater than* 1, so secant is never *less than* 1 in absolute value. When the graph of sine crosses the x axis, an asymptote for a matching graph of cosecant will appear there. 

The general form of cosecant is:
$$ y = A\csc(B{x} - C) + D $$
$A$, $B$, $C$, and $D$ will have similar meanings to the cosecant function as they did to the sine and cosine functions. 

# Features of Secant and Cosecant
- The stretching factor is $|A|$
- The period is $\frac{2\pi}{|B|}$
- The domain of secant is all $x$, where $x \ne \frac{C}{B} + \frac{\pi}{2} + \frac{\pi}{|B|}k$, where $k$ is an integer. (Every half period + phase shift is where asymptotes appear)
- The domain of cosecant is all $x$, where $x \ne \frac{C}{B} + \frac{\pi}{|B|}k$, where $k$ is an integer.
- The range is $(\infty, -|A| +D]\cup [|A| + D], \infty)$
- The vertical asymptotes of secant occur at $x = \frac{C}{B} + \frac{\pi}{2} + \frac{\pi}{2} + \frac{\pi}{|B|}k$, where $k$ is an integer.
- The vertical asymptotes of cosecant occur at $x = \frac{C}{B} + \frac{\pi}{|B|}k$, where $k$ is an integer.
- The vertical shift is $D$. 
# Inverse Functions
For any one to one function $f(x) = y$, a function $f^{-1}(y) = x)$. A function is considered one-to-one if every input only has one output, and every output can only be created from a single input. 

The inverse of a trig function is denoted as $sin^{-1}$, or $arcsin$ respectively. 

The inverse of a trig function is **not** the same as the reciprocal of a trig function, $\frac{1}{sin}$ is not the same as $sin^{-1}$. 

- The *domain* of $f$ is the *range* of $f^{-1}$. 
- The *range* of $f$ is the *domain* of $f^{-1}$. 

| Trig functions                      | Inverse trig functions               |
| ----------------------------------- | ------------------------------------ |
| Domain: Angle measures              | Domain: Ratio of sides of a triangle |
| Range: Ratio of sides of a triangle | Range: Angle Measure                 |
- To find the inverse of sin, you need to restrict the domain to $[-\frac{\pi}{2}, \frac{\pi}{2}]$
- To find the inverse of cos, you need to restrict the domain to $[0, \pi]$
- To find the inverse of tangent, you need to restrict the domain to $(-\frac{\pi}{2}, \frac{\pi}{2})$.

The graphs of an inverse function can be found by taking the graph of $f$, and flipping it over the line $y=x$. 

# Examples
> Given $-2\tan(\pi*x + \pi) - 1$

 $A = -2$, $B = \pi$, $C = -\pi$, $D = -1$

> Identify the vertical stretch/compress factor, period, phase shift, and vertical shift of the function $y = 4\sec(\frac{\pi}{3}x - \frac{\pi}{2}) + 1$ 

$A = 4$, $B = \frac{\pi}{3}$, $C = \frac{\pi}{2}$, $D = 4$

Vertical stretch: $|4| = 4$
Period: $\frac{2\pi}{\frac{\pi}{3}} = \frac{2\pi}{1} * \frac{3}{\pi} = 6$
Phase shift: $\dfrac{\frac{\pi}{2}}{\frac{\pi}{3}} = \frac{3}{2}$
Vertical shift: $1$

| Transformation | Equation                  |
| -------------- | ------------------------- |
| Stretch        | $\|-2\| = 2$              |
| Period         | $\frac{\pi}{\|\pi\|} = 1$ |
| Phase shift    | $\frac{-\pi}{\pi} = -1$   |
| Vertical shift | $-1$                      |
> Evaluate $\arccos{\frac{1}{2}}$ using the unit circle. 

Taking the inverse of the above function, we get this. Because the domain of $cos$ ranges from $0$ to $\pi$ inclusive, the answer is going to be in quadrant 1 or quadrant 2. 
$$ cos(a) = \frac{1}{2} $$
When $x$ is equal to one half, the angle is equal to $\frac{\pi}{3}$.