Mathematics
Algebra
Function Composition And Notation Conveniences
Right to Left Evaluation
$$\begin{equation}
\begin{split}
f \triangleleft x &= f(x)\\
f \triangleleft g \triangleleft x &= f(g(x))\\
f \circ g \triangleleft x &= f(g(x))
\end{split}
\end{equation}$$
(Reading and evaluating expressions from right to left should be the
standard in mathematics, as it is in Haskell.)
Left to Right Evaluation
$$\begin{equation}
\begin{split}
x \triangleright f &= f(x)\\
x \triangleright f \triangleright g &= g(f(x))
\end{split}
\end{equation}$$
(Somewhat confusing since it's not right to left,
but can be convenient and look nice...)
Parentheses are evil, inelegant, and requires more work to write/type and read.
The fewer parentheses the better. When things get complicated, smart use of function composition
on the other hand looks beautiful when used well.
Algebra Basics
$$\begin{equation}
\begin{split}
a^{-x} &= \frac{1}{a^x}
\end{split}
\end{equation}$$
$$\begin{equation}
\begin{split}
a^x a^y &= a^{\left(x + y\right)}
\end{split}
\end{equation}$$
$$\begin{equation}
\begin{split}
\frac{a^x}{a^y} &= a^{\left(x - y\right)}
\end{split}
\end{equation}$$
$$\begin{equation}
\begin{split}
\left(a^x\right)^y &= a^{\left(x y\right)}
\end{split}
\end{equation}$$
$$\begin{equation}
\begin{split}
a^0 &= 1 \\
a^1 &= a \\
a^{\frac{1}{n}} &= \sqrt[n]{a}
\end{split}
\end{equation}$$
$$\begin{equation}
\begin{split}
\frac{1}{\frac{a}{b}} &= \left(\frac{a}{b}\right)^{-1} &= \frac{b}{a}
\end{split}
\end{equation}$$
$$\begin{equation}
\begin{split}
a &= e^x \\
\ln a &= x
\end{split}
\end{equation}$$
$$\begin{equation}
\begin{split}
\ln \triangleleft\; e^x &= x \\
e^{\ln{\left(x\right)}} &= x
\end{split}
\end{equation}$$
$$\begin{equation}
\begin{split}
\ln \triangleleft\; a \cdot b &= \ln{a} + \ln{b}
\end{split}
\end{equation}$$
$$\begin{equation}
\begin{split}
\ln \triangleleft\; \frac{a}{b} &= \ln{a} - \ln{b}
\end{split}
\end{equation}$$
$$\begin{equation}
\begin{split}
\ln \triangleleft\; a^n &= n \cdot \ln{a}
\end{split}
\end{equation}$$
Trigonometry
The Unit Circle & Special Angles In Trig
Warning
Never use Pi (\(\pi\))! It makes (thinking in terms of) radians confusing,
Tao (\(\tau\)) is what the enlightened trigonometer uses,
and won't screw you over.
To easily memorize the special angles in trig, notice the repeating patterns on the above angles.
- For values on the x-axis, anything over \(\frac{1}{4}\tau\) and under \(\frac{3}{4}\tau\) will be negative
- For values on the y-axis, anything over \(\frac{1}{2}\tau\) will be negative
-
Diagonals will be \(\pm \frac{\sqrt{2}}{2}\)
For ratios of \(\frac{1}{12}\tau\) on the sides, i.e.
\(\frac{1}{12}\tau\),
\(\frac{2}{12}\tau\),
\(\frac{4}{12}\tau\),
\(\frac{5}{12}\tau\),
\(\frac{7}{12}\tau\),
\(\frac{8}{12}\tau\),
\(\frac{10}{12}\tau\), and
\(\frac{11}{12}\tau\). Draw a circle and dot the point where it occurs
(which is pretty easy since the above are simple ratios of a circle
when expressed in terms of \(\tau\)).
Then with regards to the \(x\) and \(y\) axis values:
- The longer size will be \(\pm \frac{\sqrt{3}}{2}\)
- The shorter side will be \(\pm \frac{1}{2}\)
See the above examples.
Conventions
$$\begin{equation}
\begin{split}
\sin\theta &= \frac{b}{c}\\
\cos\theta &= \frac{a}{c}\\
\tan\theta &= \frac{b}{a}
\end{split}
\end{equation}$$
$$\begin{equation}
\begin{split}
\csc\theta &= \frac{1}{\sin\theta} = \frac{c}{b}\\
\sec\theta &= \frac{1}{\cos\theta} = \frac{c}{b}\\
\cot\theta &= \frac{1}{\tan\theta} = \frac{a}{b}
\end{split}
\end{equation}$$
Radians & Radian Conversion
Constants
$$\begin{equation}
\begin{split}
\tau &= 2\pi = 360^{\circ} \\
\pi &= \frac{1}{2}\tau = 180^{\circ}
\end{split}
\end{equation}$$
Conversion
Given
$$\begin{equation}
\begin{split}
{1}^{\circ}
&= \frac{{1}}{360} \tau \; {\displaystyle {\mathrm{rad}}} \\
{1} \; \mathrm{rad}
&= \frac{{1}}{\tau} \cdot 360^{\circ}
= {1} \cdot \frac{360^{\circ}}{\tau}
\end{split}
\end{equation}$$
Degrees to Radians (in terms of \(\tau\))
$$\begin{equation}
\begin{split}
{x^{\circ}} &= \frac{{x}}{360} \tau \; {\displaystyle {\mathrm{rad}}}
\end{split}
\end{equation}$$
Radians to (terrible and arbitrary) Degrees (the formula doesn't even look as nice)
$$\begin{equation}
\begin{split}
{x} \; \mathrm{rad}
&= \frac{{x}}{\tau} \cdot 360^{\circ} \\
&= {x} \cdot \frac{360^{\circ}}{\tau} \\
&= {x} \cdot \frac{180^{\circ}}{\pi}
\end{split}
\end{equation}$$
Geometry
The Circle
$$\begin{equation}
\begin{split}
\small\text{Area}\;\normalsize &= \pi r^2\\
\small\text{Circumference}\;\normalsize &= 2 \pi r
\end{split}
\end{equation}$$
Pre-Calculus
Calculus
Function Graphs
$$\begin{equation}
\begin{split}
\frac{\mathrm{d}}{\mathrm{d}x} x^n &= n \cdot x^{n -1}
\end{split}
\end{equation}$$
$$\begin{equation}
\begin{split}
\int 1 \;\mathrm{d}x &= x + \mathrm{c}
\end{split}
\end{equation}$$
$$\begin{equation}
\begin{split}
\int a \;\mathrm{d}x &= ax + \mathrm{c}
\end{split}
\end{equation}$$
$$\begin{equation}
\begin{split}
\int x^n \;\mathrm{d}x &= \frac{x^{x + 1}}{n + 1} + \mathrm{c}
\end{split}
\end{equation}$$
