Trig formulas

Memorizing these were a pain

This is a quick reference sheet for sets of trig formulas whose members I had a hard time memorizing.

Trig identities

You'll have to memorize the sum and difference formulas, but you can derive the double angle formulas, and the half angle formulas will be given when needed.

Sum and difference

\def \+ {\textcolor{ffa5a5}{+}} \def \- {\textcolor{a5b8ff}{-}} \def \ccos {\textcolor{ffcb7e}{\cos}} \def \csin {\textcolor{fffb7e}{\sin}} \def \ctan {\textcolor{cff4ff}{\tan}} \begin{aligned} \ccos \left(A \- B\right) &= \ccos A \ccos B \+ \csin A \csin B \\[1ex] \ccos \left(A \+ B\right) &= \ccos A \ccos B \- \csin A \csin B \\[1ex] \csin \left(A \- B\right) &= \csin A \ccos B \- \ccos A \csin B \\[1ex] \csin \left(A \+ B\right) &= \csin A \ccos B \+ \ccos A \csin B \\[1ex] \ctan \left(A \- B\right) &= \dfrac{\ctan A \- \ctan B}{1 \+ \ctan A \ctan B} \\[3ex] \ctan \left(A \+ B\right) &= \dfrac{\ctan A \+ \ctan B}{1 \- \ctan A \ctan B} \end{aligned}

Double angle

\begin{aligned} \sin 2x &= 2 \sin x \cos x \\[2ex] \cos 2x &= \cos^2 x - \sin^2 x \\ &= 1 - 2 \sin^2 x \\ &= 2 \cos^2 x - 1 \\[2ex] \tan 2x &= \dfrac{2 \tan x}{1 - \tan^2 x} \end{aligned}

Half angle

\begin{aligned} \sin\dfrac{\theta}{2} &= \pm\sqrt{\dfrac{1 - \cos \theta}{2}} \\[2ex] \cos\dfrac{\theta}{2} &= \pm\sqrt{\dfrac{1 + \cos \theta}{2}} \\[2ex] \tan \dfrac{\theta}{2} &= \pm\sqrt{\dfrac{1 - \cos \theta}{1 + \cos \theta}} \\[2ex] &= \dfrac{\sin \theta}{1 + \cos \theta} \\[2ex] &= \dfrac{1 - \cos \theta}{\sin \theta} \end{aligned}

Calculus

You need to memorize the derivatives and the integrals for sine and cosine, but you probably don't need to know the integrals for secant and cosecant.

Derivatives

\def \- {\textcolor{ff8585}{-}} \gdef \fun #1 {\textcolor{a8ebff}{#1}} \gdef \cfn #1 {\textcolor{ffcb7e}{#1}} \begin{aligned} \dfrac{d}{dx} \fun \sin x &= \cfn \cos x & \dfrac{d}{dx} \cfn \cos x &= \-\fun \sin x \\[2ex] \dfrac{d}{dx} \fun \tan x &= \fun \sec^2 x & \dfrac{d}{dx} \cfn \cot x &= \-\cfn \csc^2 x \\[2ex] \dfrac{d}{dx} \fun \sec x &= \fun \sec x \fun \tan x & \dfrac{d}{dx} \cfn \csc x &= \-\cfn \csc x \cfn \cot x \end{aligned}

Derivatives of inverses

\def \- {\textcolor{ff8585}{-}} \begin{aligned} \dfrac{d}{dx} \sin^{-1} x &= \dfrac{1}{\sqrt{1 - x^2}} & \dfrac{d}{dx} \cos^{-1} x &= \-\dfrac{1}{\sqrt{1 - x^2}} \\[2ex] \dfrac{d}{dx} \tan^{-1} x &= \dfrac{1}{1 + x^2} & \dfrac{d}{dx} \cot^{-1} x &= \-\dfrac{1}{1 + x^2} \\[2ex] \dfrac{d}{dx} \sec^{-1} x &= \dfrac{1}{\left|x\right| \sqrt{x^2 - 1}} & \dfrac{d}{dx} \csc^{-1} x &= \-\dfrac{1}{\left|x\right| \sqrt{x^2 - 1}} \end{aligned}

Integrals

\def \integral {\int\!} \def \dx {\,dx} \def \c {&\textcolor{595E63}{+ \; C}} \begin{aligned} \integral \sin x \dx &= -\cos x \c \\ \integral \cos x \dx &= \sin x \c \\ \integral \tan x \dx &= -\ln\left|\cos x\right| \c \\ &= \ln\left|\sec x\right| \c \\ \integral \cot x \dx &= \ln\left|\sin x\right| \c \\ \integral \sec x \dx &= \ln\left|\sec x + \tan x\right| \c \\ \integral \csc x \dx &= -\ln\left|\csc x + \cot x\right| \c \end{aligned}

Other painful formulas

I believe you need to know all of these.

Taylor series

\def \- {\textcolor{ff8585}{-}} \def \negOne {\textcolor{ff8585}{\left( -1 \right) ^ n}} \def \! {\textcolor{73d471}{!}} \gdef \expanded #1 {\textcolor{8e97a0}{#1}} \begin{aligned} \sin x &= \expanded {x \- \dfrac{x^3}{3\!} + \dfrac{x^5}{5\!} \- \dfrac{x^7}{7\!} + \dots} = \sum_{n=0}^{\infty} \dfrac{\negOne x ^ {2n + 1}}{\left( 2n + 1 \right)\!} \\[3ex] \cos x &= \expanded {1 \- \dfrac{x^2}{2\!} + \dfrac{x^4}{4\!} \- \dfrac{x^6}{6\!} + \dots} = \sum_{n=0}^{\infty} \dfrac{\negOne x ^ {2n}}{\left( 2n \right)\!} \\[3ex] e^x &= \expanded {1 + x + \dfrac{x^2}{2\!} + \dfrac{x^3}{3\!} + \dfrac{x^4}{4\!} + \dots} = \sum_{n=0}^{\infty} \dfrac{x^n}{n\!} \\[3ex] \dfrac{1}{1 - x} &= \expanded {1 + x + x^2 + x^3 + x^4 + \dots} = \sum_{n=0}^{\infty} x^n \\[3ex] \tan^{-1} x &= \expanded {x \- \dfrac{x^3}{3} + \dfrac{x^5}{5} \- \dfrac{x^7}{7} + \dots} = \sum_{n=0}^{\infty} \dfrac{\negOne x ^ {2n + 1}}{2n + 1} \\[3ex] \ln x &= \expanded {\left(x - 1\right) \- \dfrac{\left(x - 1\right)^2}{2} + \dfrac{\left(x - 1\right)^3}{3} + \dots} = \sum_{n=1}^{\infty} \dfrac{\textcolor{ff8585}{\left( -1 \right) ^ {n + 1}} \left(x - 1\right)}{n} \end{aligned}

Lagrange form of the error

Typically used only on non-alternating series.

R_n < \left| \dfrac{M}{\left( n + 1 \right)!} \left( x - a \right)^{n+1} \right| \\[1ex]