$
% macros
% MathJax
\newcommand{\bigtimes}{\mathop{\vcenter{\hbox{$\Huge\times\normalsize$}}}} % prefix with \! and postfix with \!\!
% sized grouping symbols
\renewcommand {\aa} [1] {\left\langle {#1} \right\rangle} % <> angle brackets
\newcommand {\bb} [1] {\left[ {#1} \right]} % [] brackets
\newcommand {\cc} [1] {\left\{ {#1} \right\}} % {} curly braces
\newcommand {\mm} [1] {\lVert {#1} \rVert} % || || double norm bars
\newcommand {\nn} [1] {\lvert {#1} \rvert} % || norm bars
\newcommand {\pp} [1] {\left( {#1} \right)} % () parentheses
% unit
\newcommand {\unit} [1] {\bb{\mathrm{#1}}} % measurement unit
% math
\newcommand {\fn} [1] {\mathrm{#1}} % function name
% sets
\newcommand {\setZ} {\mathbb{Z}}
\newcommand {\setQ} {\mathbb{Q}}
\newcommand {\setR} {\mathbb{R}}
\newcommand {\setC} {\mathbb{C}}
% arithmetic
\newcommand {\q} [2] {\frac{#1}{#2}} % quotient, because fuck \frac
% trig
\newcommand {\acos} {\mathrm{acos}} % \mathrm{???}^{-1} is misleading
\newcommand {\asin} {\mathrm{asin}}
\newcommand {\atan} {\mathrm{atan}}
\newcommand {\atantwo} {\mathrm{atan2}} % at angle = atan2(y, x)
\newcommand {\asec} {\mathrm{asec}}
\newcommand {\acsc} {\mathrm{acsc}}
\newcommand {\acot} {\mathrm{acot}}
% complex numbers
\newcommand {\z} [1] {\tilde{#1}}
\newcommand {\conj} [1] {{#1}^\ast}
\renewcommand {\Re} {\mathfrak{Re}} % real part
\renewcommand {\Im} {\mathrm{I}\mathfrak{m}} % imaginary part
% quaternions
\newcommand {\quat} [1] {\tilde{\mathbf{#1}}} % quaternion symbol
\newcommand {\uquat} [1] {\check{\mathbf{#1}}} % versor symbol
\newcommand {\gquat} [1] {\tilde{\boldsymbol{#1}}} % greek quaternion symbol
\newcommand {\guquat}[1] {\check{\boldsymbol{#1}}} % greek versor symbol
% vectors
\renewcommand {\vec} [1] {\mathbf{#1}} % vector symbol
\newcommand {\uvec} [1] {\hat{\mathbf{#1}}} % unit vector symbol
\newcommand {\gvec} [1] {\boldsymbol{#1}} % greek vector symbol
\newcommand {\guvec} [1] {\hat{\boldsymbol{#1}}} % greek unit vector symbol
% special math vectors
\renewcommand {\r} {\vec{r}} % r vector [m]
\newcommand {\R} {\vec{R}} % R = r - r' difference vector [m]
\newcommand {\ur} {\uvec{r}} % r unit vector [#]
\newcommand {\uR} {\uvec{R}} % R unit vector [#]
\newcommand {\ux} {\uvec{x}} % x unit vector [#]
\newcommand {\uy} {\uvec{y}} % y unit vector [#]
\newcommand {\uz} {\uvec{z}} % z unit vector [#]
\newcommand {\urho} {\guvec{\rho}} % rho unit vector [#]
\newcommand {\utheta} {\guvec{\theta}} % theta unit vector [#]
\newcommand {\uphi} {\guvec{\phi}} % phi unit vector [#]
\newcommand {\un} {\uvec{n}} % unit normal vector [#]
% vector operations
\newcommand {\inner} [2] {\left\langle {#1} , {#2} \right\rangle} % <,>
\newcommand {\outer} [2] {{#1} \otimes {#2}}
\newcommand {\norm} [1] {\mm{#1}}
\renewcommand {\dot} {\cdot} % dot product
\newcommand {\cross} {\times} % cross product
% matrices
\newcommand {\mat} [1] {\mathbf{#1}} % matrix symbol
\newcommand {\gmat} [1] {\boldsymbol{#1}} % greek matrix symbol
% ordinary derivatives
\newcommand {\od} [2] {\q{d #1}{d #2}} % ordinary derivative
\newcommand {\odn} [3] {\q{d^{#3}{#1}}{d{#2}^{#3}}} % nth order od
\newcommand {\odt} [1] {\q{d{#1}}{dt}} % time od
% partial derivatives
\newcommand {\de} {\partial} % partial symbol
\newcommand {\pd} [2] {\q{\de{#1}}{\de{#2}}} % partial derivative
\newcommand {\pdn} [3] {\q{\de^{#3}{#1}}{\de{#2}^{#3}}} % nth order pd
\newcommand {\pdd} [3] {\q{\de^2{#1}}{\de{#2}\de{#3}}} % 2nd order mixed pd
\newcommand {\pdt} [1] {\q{\de{#1}}{\de{t}}} % time pd
% vector derivatives
\newcommand {\del} {\nabla} % del
\newcommand {\grad} {\del} % gradient
\renewcommand {\div} {\del\dot} % divergence
\newcommand {\curl} {\del\cross} % curl
% differential vectors
\newcommand {\dL} {d\vec{L}} % differential vector length [m]
\newcommand {\dS} {d\vec{S}} % differential vector surface area [m^2]
% special functions
\newcommand {\Hn} [2] {H^{(#1)}_{#2}} % nth order Hankel function
\newcommand {\hn} [2] {H^{(#1)}_{#2}} % nth order spherical Hankel function
% transforms
\newcommand {\FT} {\mathcal{F}} % fourier transform
\newcommand {\IFT} {\FT^{-1}} % inverse fourier transform
% signal processing
\newcommand {\conv} [2] {{#1}\ast{#2}} % convolution
\newcommand {\corr} [2] {{#1}\star{#2}} % correlation
% abstract algebra
\newcommand {\lie} [1] {\mathfrak{#1}} % lie algebra
% other
\renewcommand {\d} {\delta}
% optimization
%\DeclareMathOperator* {\argmin} {arg\,min}
%\DeclareMathOperator* {\argmax} {arg\,max}
\newcommand {\argmin} {\fn{arg\,min}}
\newcommand {\argmax} {\fn{arg\,max}}
% waves
\renewcommand {\l} {\lambda} % wavelength [m]
\renewcommand {\k} {\vec{k}} % wavevector [rad/m]
\newcommand {\uk} {\uvec{k}} % unit wavevector [#]
\newcommand {\w} {\omega} % angular frequency [rad/s]
\renewcommand {\TH} {e^{j \w t}} % engineering time-harmonic function [#]
% classical mechanics
\newcommand {\F} {\vec{F}} % force [N]
\newcommand {\p} {\vec{p}} % momentum [kg m/s]
% \r % position [m], aliased
\renewcommand {\v} {\vec{v}} % velocity vector [m/s]
\renewcommand {\a} {\vec{a}} % acceleration [m/s^2]
\newcommand {\vGamma} {\gvec{\Gamma}} % torque [N m]
\renewcommand {\L} {\vec{L}} % angular momentum [kg m^2 / s]
\newcommand {\mI} {\mat{I}} % moment of inertia tensor [kg m^2/rad]
\newcommand {\vw} {\gvec{\omega}} % angular velocity vector [rad/s]
\newcommand {\valpha} {\gvec{\alpha}} % angular acceleration vector [rad/s^2]
% electromagnetics
% fields
\newcommand {\E} {\vec{E}} % electric field intensity [V/m]
\renewcommand {\H} {\vec{H}} % magnetic field intensity [A/m]
\newcommand {\D} {\vec{D}} % electric flux density [C/m^2]
\newcommand {\B} {\vec{B}} % magnetic flux density [Wb/m^2]
% potentials
\newcommand {\A} {\vec{A}} % vector potential [Wb/m], [C/m]
% \F % electric vector potential [C/m], aliased
% sources
\newcommand {\I} {\vec{I}} % line current density [A] , [V]
\newcommand {\J} {\vec{J}} % surface current density [A/m] , [V/m]
\newcommand {\K} {\vec{K}} % volume current density [A/m^2], [V/m^2]
% \M % magnetic current [V/m^2], aliased, obsolete
% materials
\newcommand {\ep} {\epsilon} % permittivity [F/m]
% \mu % permeability [H/m], aliased
\renewcommand {\P} {\vec{P}} % polarization [C/m^2], [Wb/m^2]
% \p % electric dipole moment [C m], aliased
\newcommand {\M} {\vec{M}} % magnetization [A/m], [V/m]
\newcommand {\m} {\vec{m}} % magnetic dipole moment [A m^2]
% power
\renewcommand {\S} {\vec{S}} % poynting vector [W/m^2]
\newcommand {\Sa} {\aa{\vec{S}}_t} % time averaged poynting vector [W/m^2]
% quantum mechanics
\newcommand {\bra} [1] {\left\langle {#1} \right|} % <|
\newcommand {\ket} [1] {\left| {#1} \right\rangle} % |>
\newcommand {\braket} [2] {\left\langle {#1} \middle| {#2} \right\rangle}
$
Motivation
The study of triangles.
- Angles
- Pythagorean Theorem
- Trigonometric Functions
- Inverse Trigonometric Functions
- Pythagorean Identity
- Law of Cosines
- Law of Sines
- Sum and Difference Identities
- Other Identities
- Taylor Series
- Complex Exponentials
Angles
- Radians
$$
\theta \unit{rad} \equiv \q{s}{r}
$$
$$
s_{circle} = 2 \pi r
$$
$$
\theta_{circle} = 2 \pi \unit{rad}
$$
- Turns
$$
1 \unit{turn} \equiv 2 \pi \unit{rad}
$$
- Degrees
$$
360 \unit{deg} \equiv 2 \pi \unit{rad}
$$
- Types
- zero
- acute
- right
- obtuse
- straight
- reflex
- full
Pythagorean Theorem
$$
\boxed{
c^2 = a^2 + b^2
}
$$
Trigonometric Functions
$$
\begin{align*}
\sin \theta \equiv \q{o}{h} &\quad\quad \csc \theta \equiv \q{h}{o} \\
\cos \theta \equiv \q{a}{h} &\quad\quad \sec \theta \equiv \q{h}{a} \\
\tan \theta \equiv \q{o}{a} &\quad\quad \cot \theta \equiv \q{a}{o}
\end{align*}
$$
$$
\tan \theta = \q{\sin \theta}{\cos \theta} \quad\quad
\csc \theta = \q{1}{\sin \theta} \quad\quad
\sec \theta = \q{1}{\cos \theta} \quad\quad
\cot \theta = \q{\cos \theta}{\sin \theta}
$$
$$
x = h \cos \theta \quad\quad y = h \sin \theta
$$
$$
\sin(\theta + 2 \pi) = \sin \theta
$$
$$
\sin \pp{ \q{\pi}{2} - \theta } = \cos \theta \\
\cos \pp{ \q{\pi}{2} - \theta } = \sin \theta
$$
$$
\sin(-\theta) = - \sin \theta \\
\cos(-\theta) = \cos \theta
$$
Inverse Trigonometric Functions
$$
\begin{array}{lll}
\asin(x) = \theta & x \in \bb{-1, 1} & \theta \in \bb{-\q{\pi}{2}, \q{\pi}{2}} \\
\acos(x) = \theta & x \in \bb{-1, 1} & \theta \in \bb{0, \pi} \\
\atan(x) = \theta & x \in \mathbb{R} & \theta \in \bb{-\q{\pi}{2}, \q{\pi}{2}} \\
\acsc(x) = \theta & x \in (-\infty,-1] \cup [1,\infty) & \theta \in \bb{-\q{\pi}{2}, \q{\pi}{2}} \\
\asec(x) = \theta & x \in (-\infty,-1] \cup [1,\infty) & \theta \in \bb{0, \pi} \\
\acot(x) = \theta & x \in \mathbb{R} & \theta \in \bb{0, \pi}
\end{array}
$$
$$
\begin{array}{lll}
\atan2(y, x) = \theta & x, y \in \mathbb{R} & \theta \in \bb{-\pi, \pi} \\
\end{array}
$$
Pythagorean Identity
$$
\sin^2 \theta = (\sin \theta)(\sin \theta)
$$
$$
\boxed{
1 = \cos^2 \theta + \sin^2 \theta
}
$$
Law of Cosines
$$
\begin{align*}
c^2
&= (b \cos \theta - a)^2 + (b \sin \theta - 0)^2 \\
&= b^2 \cos^2 \theta - 2 a b \cos \theta + a^2 + b^2 \sin^2 \theta \\
&= a^2 + b^2 (\cos^2 \theta + \sin^2 \theta) - 2 a b \cos \theta \\
&= a^2 + b^2 - 2 a b \cos \theta
\end{align*}
$$
$$
\boxed{
c^2 = a^2 + b^2 - 2 a b \cos \theta
}
$$
Law of Sines
$$
\begin{align*}
c \sin \beta &= b \sin \gamma \\
\q{\sin \beta}{b} &= \q{\sin \gamma}{c}
\end{align*}
$$
$$
\boxed{
\q{\sin \alpha}{a} = \q{\sin \beta}{b} = \q{\sin \gamma}{c}
}
$$
Sum and Difference Identities
$$
\boxed{
\cos (\theta \pm \phi) = \cos \theta \cos \phi \mp \sin \theta \sin \phi
}
$$
$$
\boxed{
\sin (\theta \pm \phi) = \sin \theta \cos \phi \pm \cos \theta \sin \phi
}
$$
Other Identities
- Pythagorean Identity
$$
1 + \tan^2 \theta = \sec^2 \theta \\
1 + \cot^2 \theta = \csc^2 \theta
$$
- Law of Tangents
$$
\q{a - b}{a + b} = \q{\tan\pp{\q{1}{2}(\alpha - \beta)}}{\tan\pp{\q{1}{2}(\alpha + \beta)}}
$$
- Sum and Difference
$$
\tan(\theta \pm \phi) = \q{\tan \theta \pm \tan \phi}{1 \mp \tan \theta \tan \phi}
$$
- Double Angle
$$
\sin(2 \theta) = 2 \sin \theta \cos \theta = \q{2 \tan \theta}{1 + \tan^2 \theta} \\
\cos(2 \theta) = \cos^2 \theta - \sin^2 \theta = 2 \cos^2 \theta - 1 = 1 - 2 \sin^2 \theta = \q{1 - \tan^2 \theta}{1 + \tan^2 \theta} \\
$$
- Half Angle
$$
\sin^2 \pp{ \q{\theta}{2} } = \q{1}{2}(1 - \cos \theta) \\
\cos^2 \pp{ \q{\theta}{2} } = \q{1}{2}(1 + \cos \theta)
$$
- Product-to-sum
- Sum-to-product
- Power Reduction
Taylor Series
$$
f(x) = \sum_{n=0}^{\infty} \q{1}{n!} \odn{f}{x}{n} \bigg|_{x=x_0} (x - x_0)^n
$$
$$
\cos x
= \sum_{n=0}^\infty (-1)^n \q{x^{2n}}{(2n)!}
= \q{x^0}{0!} - \q{x^2}{2!} + \q{x^4}{4!} - \q{x^6}{6!} + \ldots
$$
$$
\sin x
= \sum_{n=0}^\infty (-1)^n \q{x^{2n+1}}{(2n+1)!}
= \q{x^1}{1!} - \q{x^3}{3!} + \q{x^5}{5!} - \q{x^7}{7!} + \ldots
$$
$$
e^x
= \sum_{n=0}^{\infty} \q{x^n}{n!}
= \q{x^0}{0!} + \q{x^1}{1!} + \q{x^2}{2!} + \q{x^3}{3!} + \ldots
$$
$$
e^{i x}
= \sum_{n=0}^{\infty} \q{(i x)^n}{n!}
= \q{x^0}{0!} + i \q{x^1}{1!} - \q{x^2}{2!} - i \q{x^3}{3!} + \ldots
$$
$$
e^{i x} = \cos x + i \sin x
$$
Complex Exponential
$$
\boxed{
e^{i \theta} = \cos \theta + i \sin \theta
}
$$
$$
\sin \theta = \q{1}{i 2} \pp{ e^{i \theta} - e^{-i \theta} }
$$
$$
\cos \theta = \q{1}{2} \pp{ e^{i \theta} + e^{-i \theta} }
$$
$$
\begin{align*}
e^{i (\theta \pm \phi)}
&= e^{i \theta} e^{\pm i \phi} \\
\cos (\theta \pm \phi) + i \sin (\theta \pm \phi)
&= (\cos \theta + i \sin \theta) (\cos \phi \pm i \sin \phi) \\
\cos (\theta \pm \phi) + i \sin (\theta \pm \phi)
&= (\cos \theta \cos \phi \mp \sin \theta \sin \phi) + i (\sin \theta \cos \phi \pm \cos \theta \sin \phi)
\end{align*}
$$