Andrew's git - notes.git/blob

f30ed7adc4874089ba307799e81c327921756838
   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**Mean / equilibrium:** line that the graph oscillates around ($y=d$)
  28
  29## Solving trig equations
  30
  311. Solve domain for $n\theta$
  322. Find solutions for $n\theta$
  333. Divide solutions by $n$
  34
  35$\sin2\theta={\sqrt{3}\over2}, \quad \theta \in[0, 2\pi] \quad(\therefore 2\theta \in [0,4\pi])$
  36$2\theta=\sin^{-1}{\sqrt{3} \over 2}$
  37$2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}$
  38$\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}$
  39
  40### Amplitude
  41
  42Amplitude of $a$ means graph oscillates between $+a$ and $-a$ in $y$-axis
  43
  44$a=0$ produces straight line
  45$a\lt0$ inverts the phase ($\sin$ becomes $\cos$, vice vera)
  46
  47### Period
  48
  49Period $T$ is ${2 \pi}\over b$
  50$b=0$ produces straight line
  51$b\lt0$ inverts the phase
  52
  53### Phase
  54
  55$c$ moves the graph left-right in the $x$ axis.
  56If $c=T={{2\pi}\over b}$, the graph has no actual phase shift.
  57
  58## Symmetry
  59
  60$$\sin(\theta+{\pi\over 2})=\sin\theta$$
  61$$\sin(\theta+\pi)=-\sin\theta$$
  62
  63$$\cos(\theta+{\pi \over 2})=-\cos\theta$$
  64$$\cos(\theta+\pi)=-cos(\theta+{3\pi \over 2})=\cos(-\theta)$$
  65
  66## Pythagorean identity
  67
  68$$\cos^2\theta+\sin^2\theta=1$$
  69
  70## Complementary relationships
  71
  72$$\sin({\pi \over 2} - \theta)=\cos\theta$$
  73$$\cos({\pi \over 2} - \theta)=\sin\theta$$
  74
  75$$\sin\theta=-\cos(\theta+{\pi \over 2})$$
  76$$\cos\theta=\sin(\theta+{\pi \over 2})$$
  77
  78## $tan$ graph
  79
  80$$y=a\tan(nx)$$
  81
  82where
  83$a$ is $x$-dilation (period)
  84$n$ is $y$-dilation ($\equiv$ amplitude)
  85period $T$ is $\pi \over n$
  86range is $R$
  87roots at $x={k\pi \over n}$
  88asymptotes at $x={{(2k+1)\pi}\over 2n},\quad k \in \mathbb{Z}$
  89**Asymptotes should always have equations and arrow pointing up**