-# Circular functions
+---
+geometry: a4paper, margin=2cm
+columns: 2
+author: Andrew Lorimer
+header-includes:
+- \usepackage{setspace}
+- \usepackage{fancyhdr}
+- \usepackage{graphicx}
+- \pagestyle{fancy}
+- \fancyhead[LO,LE]{Year 12 Methods}
+- \fancyhead[CO,CE]{Andrew Lorimer}
+---
-## Radians and degrees
+\setstretch{1.2}
+\pagenumbering{gobble}
-$$1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}$$
+# Circular functions
## Exact values
+\includegraphics[width=0.2\textwidth]{./graphics/exact-values-1.png}
+\includegraphics[width=0.2\textwidth]{./graphics/exact-values-2.png}
+$$1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}$$
## $\sin$ and $\cos$ graphs
$$f(x)=a \cos(bx-c)+d$$
where
-$a$ is the amplitude
-$b$ is the $x$-dilation
-$c$ is the $y$-shift
-Period is ${2 \pi} \over b$
+- $a$ is the $y$-dilation (amplitude)
+- $b$ is the $x$-dilation (period)
+- $c$ is the $x$-shift (phase)
+- $d$ is the $y$-shift (equilibrium position)
+
+
Domain is $\mathbb{R}$
+
Range is $[-b+c, b+c]$;
Graph of $\cos(x)$ starts at $(0,1)$. Graph of $\sin(x)$ starts at $(0,0)$.
**Mean / equilibrium:** line that the graph oscillates around ($y=d$)
+### Amplitude
+
+Graph oscillates between $+a$ and $-a$ in $y$-axis
+
+$a=0$ produces straight line
+
+$a < 0$ inverts the phase ($\sin$ becomes $\cos$, vice vera)
+
+### Period
+
+Period $T$ is ${2 \pi}\over b$
+
+$b=0$ produces straight line
+
+$b<0$ inverts the phase
+
+### Phase
+
+$c$ moves the graph left-right in the $x$ axis.
+
+If $c=T={{2\pi}\over b}$, the graph has no actual phase shift.
+
+## Symmetry
+
+$$\sin(\theta+{\pi\over 2})=\sin\theta$$
+$$\sin(\theta+\pi)=-\sin\theta$$
+
+$$\cos(\theta+{\pi \over 2})=-\cos\theta$$
+$$\cos(\theta+\pi)=-cos(\theta+{3\pi \over 2})=\cos(-\theta)$$
+
+## Pythagorean identity
+
+$$\cos^2\theta+\sin^2\theta=1$$
+
+## Complementary relationships
+
+$$\sin({\pi \over 2} - \theta)=\cos\theta$$
+$$\cos({\pi \over 2} - \theta)=\sin\theta$$
+
+$$\sin\theta=-\cos(\theta+{\pi \over 2})$$
+$$\cos\theta=\sin(\theta+{\pi \over 2})$$
+
+## $\tan$ graph
+
+$$y=a\tan(nx)$$
+
+where
+
+- $a$ is $x$-dilation (period)
+- $n$ is $y$-dilation ($\equiv$ amplitude)
+- period $T$ is $\pi \over n$
+- range is $R$
+- roots at $x={k\pi \over n}$
+- asymptotes at $x={{(2k+1)\pi}\over 2n},\quad k \in \mathbb{Z}$
+
+**Asymptotes should always have equations and arrow pointing up**
+
## Solving trig equations
1. Solve domain for $n\theta$
3. Divide solutions by $n$
$\sin2\theta={\sqrt{3}\over2}, \quad \theta \in[0, 2\pi] \quad(\therefore 2\theta \in [0,4\pi])$
+
$2\theta=\sin^{-1}{\sqrt{3} \over 2}$
+
$2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}$
+
$\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}$