Trigonometry

\[ \newcommand{\dx}[1]{\mathrm{d}#1} \newcommand{\ddx}[2]{\frac{\mathrm{d}#1}{\mathrm{d}#2}} \newcommand{\dddx}[2]{\frac{\mathrm{d}^2 #1}{\mathrm{d}#2^2}} \newcommand{\dddxx}[3]{\frac{\mathrm{d}^2 #1}{\mathrm{d}#2\mathrm{d}#3}} \newcommand{\pdx}[1]{\mathrm{\partial}#1} \newcommand{\pddx}[2]{\frac{\mathrm{\partial}#1}{\mathrm{\partial}#2}} \newcommand{\pdddx}[2]{\frac{\mathrm{\partial}^2 #1}{\mathrm{\partial}#2^2}} \newcommand{\pdddxx}[3]{\frac{\mathrm{\partial}^2 #1}{\mathrm{\partial}#2\mathrm{\partial}#3}} \DeclareMathOperator{\sech}{sech} \]

Trigonometric Identities

Angle Identities

Sine Cosine Tangent
\[ \begin{align} \sin(\theta + 180^\circ) & = - \sin\theta \\ \sin(\theta + 360^\circ) & = \sin\theta \end{align} \] \[ \begin{align} \cos(\theta + 180^\circ) & = - \cos\theta \\ \cos(\theta + 360^\circ) & = \cos\theta \end{align} \] \[ \begin{align} \tan(\theta + 180^\circ) & = \tan\theta \\ \tan(\theta + 360^\circ) & = \tan\theta \end{align} \]
\[ \begin{align} \sin( 90^\circ - \theta) & = \cos\theta \\ \sin(180^\circ - \theta) & = \sin\theta \end{align} \] \[ \begin{align} \cos( 90^\circ - \theta) & = \sin\theta \\ \cos(180^\circ - \theta) & = -\cos\theta \end{align} \] \[ \begin{align} \tan( 90^\circ - \theta) & = \frac{1}{\tan\theta} \\ \tan(180^\circ - \theta) & = -\tan\theta \end{align} \]
\[ \sin(-\theta) = - \sin(\theta) \] \[ \cos(-\theta) = \cos(\theta) \] \[ \tan(-\theta) = - \tan(\theta) \]

Pythagoras' Theorem

\[ \begin{align} \sin^2 x + \cos^2 x & = 1 \\ \tan^2 x + 1 & = \sec^2 x \\ 1 + \cot^2 x & = \rm{cosec}^2 x \end{align} \]

Addition Formulae

\[ \sin \left( {A \pm B} \right) = \sin A \cos B \pm \cos A \sin B \] \[ \cos \left( {A \pm B} \right) = \cos A \cos B \mp \sin A \sin B \] \[ \tan \left( {A \pm B} \right) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \]
\[ \sin 2x = 2 \sin x \cos x \] \[ \begin{align} \cos 2x & = \cos^2 x - \sin^2x \\ \cos 2x & = 2 \cos^2 x - 1 = 1 - 2 \sin^2 x \end{align} \] \[ \tan 2x = \frac{2\tan x}{1 - \tan^2 x} \]

\[ \begin{align} \sin A + \sin B & = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \\ \sin A - \sin B & = 2 \cos \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right) \\ \cos A + \cos B & = 2 \cos \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \\ \cos A - \cos B & =-2 \sin \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right) \\ 2 \sin A \sin B & = \cos \left( {A - B} \right) - \cos \left( {A + B} \right) \\ 2 \cos A \cos B & = \cos \left( {A - B} \right) + \cos \left( {A + B} \right) \\ 2 \sin A \cos B & = \sin \left( {A - B} \right) + \sin \left( {A + B} \right) \end{align} \]

Complex Identities

\[ \begin{aligned} \cos{x} & = \mathrm{Re}\{e^{ix}\} = \frac{e^{ix} + e^{-ix}}{2} \\ \sin{x} & = \mathrm{Im}\{e^{ix}\} = \frac{e^{ix} - e^{-ix}}{2i} = -i\left(\frac{e^{ix} - e^{-ix}}{2}\right) \\ \end{aligned} \]

Integral Identities

Orthogonality Conditions

From \(-L\) to \(L\): \[ \begin{aligned} \int_{-L}^{L}{\sin(mx)\sin(nx) \ \dx{x}} & = L\delta_{mn} \\ \int_{-L}^{L}{\cos(mx)\cos(nx) \ \dx{x}} & = L\delta_{mn} \\ \int_{-L}^{L}{\sin(mx)\cos(nx) \ \dx{x}} & = 0 \end{aligned} \] \[ \delta_{mn} = \begin{cases} 1 & m = n, \\ 0 & m \neq n \end{cases} \]

From \(0\) to \(L\): \[ \begin{aligned} \int_0^{L}{\sin(n\theta)\sin(m\theta) \ \dx{\theta}} \; &= \begin{cases} \frac{1}{2}L & \mathrm{if} \ n = m \\ 0 & \mathrm{if} \ n \neq m \\ \end{cases} \\ \int_0^{L}{\cos(n\theta)\cos(m\theta) \ \dx{\theta}} \; &= \begin{cases} L & \mathrm{if} \ n = m = 0 \\ \frac{1}{2}L & \mathrm{if} \ n = m \neq 0 \\ 0 & \mathrm{if} \ n \neq m \\ \end{cases} \end{aligned} \]