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

   1# Circular functions
   2
   3## Radians and degrees
   4
   5$$1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}$$
   6
   7## Exact values
   8
   9## $\sin$ and $\cos$ graphs
  10
  11$$f(x)=a \sin(bx-c)+d$$
  12$$f(x)=a \cos(bx-c)+d$$
  13
  14where
  15$a$ is the $y$-dilation (amplitude)
  16$b$ is the $x$-dilation (period)
  17$c$ is the $x$-shift (phase)
  18$d$ is the $y$-shift (equilibrium position)
  19
  20Domain is $\mathbb{R}$
  21Range is $[-b+c, b+c]$;
  22
  23Graph of $\cos(x)$ starts at $(0,1)$. Graph of $\sin(x)$ starts at $(0,0)$.
  24
  25### Amplitude
  26
  27Amplitude of $a$ means graph oscillates between $+a$ and $-a$ in $y$-axis
  28
  29$a=0$ produces straight line
  30$a\lt0$ inverts the phase ($\sin$ becomes $\cos$, vice vera)
  31
  32### Period
  33
  34Period $T$ is ${2 \pi}\over b$
  35$b=0$ produces straight line
  36$b\lt0$ inverts the phase
  37
  38### Phase
  39
  40$c$ moves the graph left-right in the $x$ axis.
  41If $c=T={{2\pi}\over b}$, the graph has no actual phase shift.
  42
  43## Symmetry
  44
  45$$\sin(\theta+{\pi\over 2})=\sin\theta$$
  46$$\sin(\theta+\pi)=-\sin\theta$$
  47
  48$$\cos(\theta+{\pi \over 2})=-\cos\theta$$
  49$$\cos(\theta+\pi)=-cos(\theta+{3\pi \over 2})=\cos(-\theta)$$
  50
  51## Pythagorean identity
  52
  53$$\cos^2\theta+\sin^2\theta=1$$
  54
  55## Complementary relationships
  56
  57$$\sin({\pi \over 2} - \theta)=\cos\theta$$
  58$$\cos({\pi \over 2} - \theta)=\sin\theta$$
  59
  60$$\sin\theta=-\cos(\theta+{\pi \over 2})$$
  61$$\cos\theta=\sin(\theta+{\pi \over 2})$$
  62
  63## $tan$ graph
  64
  65$$y=a\tan(nx)$$
  66
  67where
  68$a$ is $x$-dilation (period)
  69$n$ is $y$-dilation ($\equiv$ amplitude)
  70period $T$ is $\pi \over n$
  71range is $R$
  72roots at $x={k\pi \over n}$
  73asymptotes at $x={{(2k+1)\pi}\over 2},\quad k \in \mathbb{Z}$