Andrew's git - notes.git/blob - methods/circ-functions.md

   1---
   2geometry: a4paper, margin=2cm
   3columns: 2
   4author: Andrew Lorimer
   5header-includes:
   6- \usepackage{setspace}
   7- \usepackage{fancyhdr}
   8- \usepackage{graphicx}
   9- \pagestyle{fancy}
  10- \fancyhead[LO,LE]{Year 12 Methods}
  11- \fancyhead[CO,CE]{Andrew Lorimer}
  12---
  13
  14\setstretch{1.2}
  15\pagenumbering{gobble}
  16
  17# Circular functions
  18
  19## Exact values
  20
  21\includegraphics[width=0.2\textwidth]{./graphics/exact-values-1.png}
  22\includegraphics[width=0.2\textwidth]{./graphics/exact-values-2.png}
  23
  24$$1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}$$
  25
  26## $\sin$ and $\cos$ graphs
  27
  28$$f(x)=a \sin(bx-c)+d$$
  29$$f(x)=a \cos(bx-c)+d$$
  30
  31where
  32
  33- $a$ is the $y$-dilation (amplitude)
  34- $b$ is the $x$-dilation (period)
  35- $c$ is the $x$-shift (phase)
  36- $d$ is the $y$-shift (equilibrium position)
  37
  38
  39Domain is $\mathbb{R}$
  40
  41Range is $[-b+c, b+c]$;
  42
  43Graph of $\cos(x)$ starts at $(0,1)$. Graph of $\sin(x)$ starts at $(0,0)$.
  44
  45**Mean / equilibrium:** line that the graph oscillates around ($y=d$)
  46
  47### Amplitude
  48
  49Graph oscillates between $+a$ and $-a$ in $y$-axis
  50
  51$a=0$ produces straight line
  52
  53$a < 0$ inverts the phase ($\sin$ becomes $\cos$, vice vera)
  54
  55### Period
  56
  57Period $T$ is ${2 \pi}\over b$
  58
  59$b=0$ produces straight line
  60
  61$b<0$ inverts the phase
  62
  63### Phase
  64
  65$c$ moves the graph left-right in the $x$ axis.
  66
  67If $c=T={{2\pi}\over b}$, the graph has no actual phase shift.
  68
  69## Symmetry
  70
  71$$\sin(\theta+{\pi\over 2})=\sin\theta$$
  72$$\sin(\theta+\pi)=-\sin\theta$$
  73
  74$$\cos(\theta+{\pi \over 2})=-\cos\theta$$
  75$$\cos(\theta+\pi)=-cos(\theta+{3\pi \over 2})=\cos(-\theta)$$
  76
  77## Pythagorean identity
  78
  79$$\cos^2\theta+\sin^2\theta=1$$
  80
  81## Complementary relationships
  82
  83$$\sin({\pi \over 2} - \theta)=\cos\theta$$
  84$$\cos({\pi \over 2} - \theta)=\sin\theta$$
  85
  86$$\sin\theta=-\cos(\theta+{\pi \over 2})$$
  87$$\cos\theta=\sin(\theta+{\pi \over 2})$$
  88
  89## $\tan$ graph
  90
  91$$y=a\tan(nx)$$
  92
  93where
  94
  95- $a$ is $x$-dilation (period)
  96- $n$ is $y$-dilation ($\equiv$ amplitude)
  97- period $T$ is $\pi \over n$
  98- range is $R$
  99- roots at $x={k\pi \over n}$
 100- asymptotes at $x={{(2k+1)\pi}\over 2n},\quad k \in \mathbb{Z}$
 101
 102**Asymptotes should always have equations and arrow pointing up**
 103
 104## Solving trig equations
 105
 1061. Solve domain for $n\theta$
 1072. Find solutions for $n\theta$
 1083. Divide solutions by $n$
 109
 110$\sin2\theta={\sqrt{3}\over2}, \quad \theta \in[0, 2\pi] \quad(\therefore 2\theta \in [0,4\pi])$
 111
 112$2\theta=\sin^{-1}{\sqrt{3} \over 2}$
 113
 114$2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}$
 115
 116$\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}$