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

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   6# Circular functions
   7
   8<!-- ## Radians and degrees -->
   9
  10<!-- $$1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}$$ -->
  11
  12## Exact values
  13
  14\includegraphics[width=0.2\textwidth]{./graphics/exact-values-1.png}
  15\includegraphics[width=0.2\textwidth]{./graphics/exact-values-2.png}
  16
  17<!-- ![diag](graphics/exact-values-1.png) -->
  18<!-- ![diag](graphics/exact-values-2.png) -->
  19
  20## $\sin$ and $\cos$ graphs
  21
  22$$f(x)=a \sin(bx-c)+d$$
  23$$f(x)=a \cos(bx-c)+d$$
  24
  25where
  26
  27- $a$ is the $y$-dilation (amplitude)
  28- $b$ is the $x$-dilation (period)
  29- $c$ is the $x$-shift (phase)
  30- $d$ is the $y$-shift (equilibrium position)
  31
  32
  33Domain is $\mathbb{R}$
  34
  35Range is $[-b+c, b+c]$;
  36
  37Graph of $\cos(x)$ starts at $(0,1)$. Graph of $\sin(x)$ starts at $(0,0)$.
  38
  39**Mean / equilibrium:** line that the graph oscillates around ($y=d$)
  40
  41### Amplitude
  42
  43Graph oscillates between $+a$ and $-a$ in $y$-axis
  44
  45$a=0$ produces straight line
  46
  47$a < 0$ inverts the phase ($\sin$ becomes $\cos$, vice vera)
  48
  49### Period
  50
  51Period $T$ is ${2 \pi}\over b$
  52
  53$b=0$ produces straight line
  54
  55$b<0$ inverts the phase
  56
  57### Phase
  58
  59$c$ moves the graph left-right in the $x$ axis.
  60
  61If $c=T={{2\pi}\over b}$, the graph has no actual phase shift.
  62
  63## Symmetry
  64
  65$$\sin(\theta+{\pi\over 2})=\sin\theta$$
  66$$\sin(\theta+\pi)=-\sin\theta$$
  67
  68$$\cos(\theta+{\pi \over 2})=-\cos\theta$$
  69$$\cos(\theta+\pi)=-cos(\theta+{3\pi \over 2})=\cos(-\theta)$$
  70
  71## Pythagorean identity
  72
  73$$\cos^2\theta+\sin^2\theta=1$$
  74
  75## Complementary relationships
  76
  77$$\sin({\pi \over 2} - \theta)=\cos\theta$$
  78$$\cos({\pi \over 2} - \theta)=\sin\theta$$
  79
  80$$\sin\theta=-\cos(\theta+{\pi \over 2})$$
  81$$\cos\theta=\sin(\theta+{\pi \over 2})$$
  82
  83## $\tan$ graph
  84
  85$$y=a\tan(nx)$$
  86
  87where
  88
  89- $a$ is $x$-dilation (period)
  90- $n$ is $y$-dilation ($\equiv$ amplitude)
  91- period $T$ is $\pi \over n$
  92- range is $R$
  93- roots at $x={k\pi \over n}$
  94- asymptotes at $x={{(2k+1)\pi}\over 2n},\quad k \in \mathbb{Z}$
  95
  96**Asymptotes should always have equations and arrow pointing up**
  97
  98## Solving trig equations
  99
 1001. Solve domain for $n\theta$
 1012. Find solutions for $n\theta$
 1023. Divide solutions by $n$
 103
 104$\sin2\theta={\sqrt{3}\over2}, \quad \theta \in[0, 2\pi] \quad(\therefore 2\theta \in [0,4\pi])$
 105
 106$2\theta=\sin^{-1}{\sqrt{3} \over 2}$
 107
 108$2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}$
 109
 110$\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}$