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
  10
  11## $\sin$ and $\cos$ graphs
  12
  13$$f(x)=a \sin(bx-c)+d$$
  14$$f(x)=a \cos(bx-c)+d$$
  15
  16where
  17$a$ is the $y$-dilation (amplitude)
  18$b$ is the $x$-dilation (period)
  19$c$ is the $x$-shift (phase)
  20$d$ is the $y$-shift (equilibrium position)
  21
  22Domain is $\mathbb{R}$
  23Range is $[-b+c, b+c]$;
  24
  25Graph of $\cos(x)$ starts at $(0,1)$. Graph of $\sin(x)$ starts at $(0,0)$.
  26
  27<<<<<<< HEAD
  28**Mean / equilibrium:** line that the graph oscillates around ($y=d$)
  29
  30## Solving trig equations
  31
  321. Solve domain for $n\theta$
  332. Find solutions for $n\theta$
  343. Divide solutions by $n$
  35
  36$\sin2\theta={\sqrt{3}\over2}, \quad \theta \in[0, 2\pi] \quad(\therefore 2\theta \in [0,4\pi])$
  37$2\theta=\sin^{-1}{\sqrt{3} \over 2}$
  38$2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}$
  39$\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}$
  40=======
  41### Amplitude
  42
  43Amplitude of $a$ means graph oscillates between $+a$ and $-a$ in $y$-axis
  44
  45$a=0$ produces straight line
  46$a\lt0$ inverts the phase ($\sin$ becomes $\cos$, vice vera)
  47
  48### Period
  49
  50Period $T$ is ${2 \pi}\over b$
  51$b=0$ produces straight line
  52$b\lt0$ inverts the phase
  53
  54### Phase
  55
  56$c$ moves the graph left-right in the $x$ axis.
  57If $c=T={{2\pi}\over b}$, the graph has no actual phase shift.
  58
  59## Symmetry
  60
  61$$\sin(\theta+{\pi\over 2})=\sin\theta$$
  62$$\sin(\theta+\pi)=-\sin\theta$$
  63
  64$$\cos(\theta+{\pi \over 2})=-\cos\theta$$
  65$$\cos(\theta+\pi)=-cos(\theta+{3\pi \over 2})=\cos(-\theta)$$
  66
  67## Pythagorean identity
  68
  69$$\cos^2\theta+\sin^2\theta=1$$
  70
  71## Complementary relationships
  72
  73$$\sin({\pi \over 2} - \theta)=\cos\theta$$
  74$$\cos({\pi \over 2} - \theta)=\sin\theta$$
  75
  76$$\sin\theta=-\cos(\theta+{\pi \over 2})$$
  77$$\cos\theta=\sin(\theta+{\pi \over 2})$$
  78
  79## $tan$ graph
  80
  81$$y=a\tan(nx)$$
  82
  83where
  84$a$ is $x$-dilation (period)
  85$n$ is $y$-dilation ($\equiv$ amplitude)
  86period $T$ is $\pi \over n$
  87range is $R$
  88roots at $x={k\pi \over n}$
  89asymptotes at $x={{(2k+1)\pi}\over 2},\quad k \in \mathbb{Z}$
  90>>>>>>> 924c0548b3e7564d4015e879c56a46a5606807fe