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?hp=1cb56c1fb7d4d43d4d519a4ad8c09378cbc1bbaf

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)
---

diff --git a/chem/water.md b/chem/water.md
index 9635ed6..3b1ccf9 100644
--- a/chem/water.md
+++ b/chem/water.md
@@ -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 --git a/methods/circ-functions.md b/methods/circ-functions.md
index a4b06c0..3bbe1c2 100644
--- a/methods/circ-functions.md
+++ b/methods/circ-functions.md
@@ -14,16 +14,17 @@ $$f(x)=a \sin(bx-c)+d$$
 $$f(x)=a \cos(bx-c)+d$$
 
 where
-$a$ is the amplitude
-$b$ is the $x$-dilation
-$c$ is the $y$-shift
+$a$ is the $y$-dilation (amplitude)
+$b$ is the $x$-dilation (period)
+$c$ is the $x$-shift (phase)
+$d$ is the $y$-shift (equilibrium position)
 
-Period is ${2 \pi} \over b$
 Domain is $\mathbb{R}$
 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
@@ -36,3 +37,54 @@ $\sin2\theta={\sqrt{3}\over2}, \quad \theta \in[0, 2\pi] \quad(\therefore 2\thet
 $2\theta=\sin^{-1}{\sqrt{3} \over 2}$
 $2\theta={\pi\over 3}, {2\pi \over 3}, {7\pi \over 3}, {8\pi \over 3}$
 $\therefore \theta = {\pi \over 6}, {\pi \over 3}, {7 \pi \over 6}, {4\pi \over 3}$
+=======
+### 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 --git a/spec/graphing.md b/spec/graphing.md
index 3e889a0..6cd013e 100644
--- a/spec/graphing.md
+++ b/spec/graphing.md
@@ -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