Trigonometry

$ % 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

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

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

$$ \boxed{ e^{i \theta} = \cos \theta + i \sin \theta } $$ $$ \sin \theta = \q{1}{i 2} \pp{ e^{i \theta} - e^{-i \theta} } $$ $$ \cos \theta = \q{1}{2} \pp{ e^{i \theta} + e^{-i \theta} } $$ $$ \begin{align*} e^{i (\theta \pm \phi)} &= e^{i \theta} e^{\pm i \phi} \\ \cos (\theta \pm \phi) + i \sin (\theta \pm \phi) &= (\cos \theta + i \sin \theta) (\cos \phi \pm i \sin \phi) \\ \cos (\theta \pm \phi) + i \sin (\theta \pm \phi) &= (\cos \theta \cos \phi \mp \sin \theta \sin \phi) + i (\sin \theta \cos \phi \pm \cos \theta \sin \phi) \end{align*} $$