From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Sun, 29 Jul 2018 11:25:36 +0000 (+1000)
Subject: Merge branch 'master' of ssh://charles/tank/andrew/school/notes
X-Git-Tag: yr11~81
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/5e2fc0ca19ac4e497696ecc4b9768da48a66e4fc?ds=inline

Merge branch 'master' of ssh://charles/tank/andrew/school/notes

locuses (spec), equilibrium for circ fn's (methods - merge w/ruslan), solvent eq's (chem)
---

5e2fc0ca19ac4e497696ecc4b9768da48a66e4fc
diff --cc chem/water.md
index 9635ed6,9635ed6..3b1ccf9
--- a/chem/water.md
+++ b/chem/water.md
@@@ -73,3 -73,3 +73,4 @@@
  ## Concentration
  - amount of solute per volume of solvent - e.g. g / L
  - relative terms - "concentrated" or "dilute"
++- mg / L = ppm = $\mu$g / g
diff --cc methods/circ-functions.md
index a4b06c0,a48b106..3bbe1c2
--- a/methods/circ-functions.md
+++ b/methods/circ-functions.md
@@@ -24,15 -22,52 +24,67 @@@ Range is $[-b+c, b+c]$
  
  Graph of $\cos(x)$ starts at $(0,1)$. Graph of $\sin(x)$ starts at $(0,0)$.
  
++<<<<<<< HEAD
 +**Mean / equilibrium:** line that the graph oscillates around ($y=d$)
 +
 +## Solving trig equations
 +
 +1. Solve domain for $n\theta$
 +2. Find solutions for $n\theta$
 +3. Divide solutions by $n$
 +
 +$\sin2\theta={\sqrt{3}\over2}, \quad \theta \in[0, 2\pi] \quad(\therefore 2\theta \in [0,4\pi])$
 +$2\theta=\sin^{-1}{\sqrt{3} \over 2}$
 +$2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}$
 +$\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}$
++=======
+ ### Amplitude
+ 
+ Amplitude of $a$ means graph oscillates between $+a$ and $-a$ in $y$-axis
+ 
+ $a=0$ produces straight line
+ $a\lt0$ inverts the phase ($\sin$ becomes $\cos$, vice vera)
+ 
+ ### Period
+ 
+ Period $T$ is ${2 \pi}\over b$
+ $b=0$ produces straight line
+ $b\lt0$ inverts the phase
+ 
+ ### Phase
+ 
+ $c$ moves the graph left-right in the $x$ axis.
+ If $c=T={{2\pi}\over b}$, the graph has no actual phase shift.
+ 
+ ## Symmetry
+ 
+ $$\sin(\theta+{\pi\over 2})=\sin\theta$$
+ $$\sin(\theta+\pi)=-\sin\theta$$
+ 
+ $$\cos(\theta+{\pi \over 2})=-\cos\theta$$
+ $$\cos(\theta+\pi)=-cos(\theta+{3\pi \over 2})=\cos(-\theta)$$
+ 
+ ## Pythagorean identity
+ 
+ $$\cos^2\theta+\sin^2\theta=1$$
+ 
+ ## Complementary relationships
+ 
+ $$\sin({\pi \over 2} - \theta)=\cos\theta$$
+ $$\cos({\pi \over 2} - \theta)=\sin\theta$$
+ 
+ $$\sin\theta=-\cos(\theta+{\pi \over 2})$$
+ $$\cos\theta=\sin(\theta+{\pi \over 2})$$
+ 
+ ## $tan$ graph
+ 
+ $$y=a\tan(nx)$$
+ 
+ where
+ $a$ is $x$-dilation (period)
+ $n$ is $y$-dilation ($\equiv$ amplitude)
+ period $T$ is $\pi \over n$
+ range is $R$
+ roots at $x={k\pi \over n}$
+ asymptotes at $x={{(2k+1)\pi}\over 2},\quad k \in \mathbb{Z}$
++>>>>>>> 924c0548b3e7564d4015e879c56a46a5606807fe
diff --cc spec/graphing.md
index 3e889a0,3e889a0..6cd013e
--- a/spec/graphing.md
+++ b/spec/graphing.md
@@@ -91,7 -91,7 +91,8 @@@ $$|(F_2P - F_1P  )| = k$
  Cartesian equation for hyperbolas ($a$ and $b$ are dilation factors):
  $${(x-h)^2 \over a^2} - {(y-k)^2 \over b^2} = 1$$
  
--Asymptotes at $y-k=\pm {b \over a}(x-h$)
++Asymptotes at $y=\pm {b \over a}(x-h)+k$
++To make hyperbola up/down rather than left/right, swap $x$ and $y$
  
  ## Parametric equations