E - Mathematical Formulas

Quadratic Formula

then

Triangle of base b and height h	Area $=\frac{1}{2}bh$
Circle of radius r	Circumference = $2\pi r$	Area = $\pi r^2$
Sphere of radius r	Surface area = $4\pi r^2$	Volume = $\frac{4}{3}\pi r^3$
Cylinder of radius r and height h	Area of curved surface = $2\pi rh$	Volume = $\pi r^2h$

Table E1 Geometry

Trigonometry

Trigonometric Identities

$\sin\theta=1/\csc\theta$
$\cos\theta=1/\sec\theta$
$\tan\theta=1/\cot\theta$
$\sin(90^{\circ}-\theta)=\cos\theta$
$\cos(90^{\circ}-\theta)=\sin\theta$
$\tan(90^{\circ}-\theta)=\cot\theta$
$\sin^2\theta+\cos^2\theta=1$
$\sec^2\theta-\tan^2\theta=1$
$\tan\theta=\sin\theta/\cos\theta$
$\sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta$
$\cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta$
$\tan(\alpha\pm\beta)=\frac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\tan\beta}$
$\sin2\theta=2\sin\theta\cos\theta$
$\cos2\theta=\cos^2\theta-\sin^2\theta=2\cos^2\theta-1=1-2\sin^2\theta$
$\sin\alpha+\sin\beta=2\sin\frac{1}{2}(\alpha+\beta)\cos\frac{1}{2}(\alpha-\beta)$
$\cos\alpha+\cos\beta=2\cos\frac{1}{2}(\alpha+\beta)\cos\frac{1}{2}(\alpha-\beta)$

Triangles

Law of sines: $\frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}$
Law of cosines: $c^2=a^2+b^2-2ab\cos\gamma$

2. Pythagorean theorem:

Series Expansions

Binomial theorem:
$(a+b)^n=a^n+na^{n-1}b+\frac{n(n-1)a^{n-2}b^2}{2!}+\frac{n(n-1)(n-2)a^{n-3}b^3}{3!}+\ldots$
$(1\pm x)^n=1\pm\frac{nx}{1!}+\frac{n(n-1)x^2}{2!}\pm\ldots\ (x^2<1)$
$(1\pm x)^{-n}=1\mp\frac{nx}{1!}+\frac{n(n-1)x^2}{2!}\mp\ldots\ (x^2<1)$
$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\ldots$
$\cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\ldots$
$\tan x=x+\frac{x^3}{3}+\frac{2x^5}{15}+\ldots$
$e^x=1+x+\frac{x^2}{2!}+\ldots$
$\ln(1+x)=x-\frac{1}{2}x^2+\frac{1}{3}x^3-\ldots\ (|x|<1)$

Derivatives

$\frac{d}{dx}[af(x)]=a\frac{d}{dx}f(x)$
$\frac{d}{dx}[f(x)+g(x)]=\frac{d}{dx}f(x)+\frac{d}{dx}g(x)$
$\frac{d}{dx}[f(x)g(x)]=f(x)\frac{d}{dx}g(x)+g(x)\frac{d}{dx}f(x)$
$\frac{d}{dx}f(u)=\left[\frac{d}{du}f(u)\right]\frac{du}{dx}$
$\frac{d}{dx}x^m=mx^{m-1}$
$\frac{d}{dx}\sin x=\cos x$
$\frac{d}{dx}\cos x=-\sin x$
$\frac{d}{dx}\tan x=\sec^2x$
$\frac{d}{dx}\cot x=-\csc^2x$
$\frac{d}{dx}\sec x=\tan x\sec x$
$\frac{d}{dx}\csc x=-\cot x\csc x$
$\frac{d}{dx}e^x=e^x$
$\frac{d}{dx}\ln x=\frac{1}{x}$
$\frac{d}{dx}\sin^{-1}x=\frac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}\cos^{-1}x=-\frac{1}{\sqrt{1-x^2}}$
$\frac{d}{dx}\tan^{-1}x=-\frac{1}{1+x^2}$

Integrals

$\int af(x)\,dx=a\int f(x)\,dx$
$\int[f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx$
$\int x^m\,dx=\frac{x^{m+1}}{m+1}\ (m\neq1)$
$~~~~~~=\ln x\ (m=-1)$
$\int\sin{x}\,dx=-\cos x$
$\int\cos{x}\,dx=\sin x$
$\int\tan{x}\,dx=\ln|\sec x|$
$\int\sin^2{ax}\,dx=\frac{x}{2}-\frac{\sin{2ax}}{4a}$
$\int\cos^2{ax}\,dx=\frac{x}{2}+\frac{\sin{2ax}}{4a}$
$\int\sin{ax}\cos{ax}\,dx=-\frac{\cos2ax}{4a}$
$\int e^{ax}\,dx=\frac{1}{a}e^{ax}$
$\int xe^{ax}\,dx=\frac{e^{ax}}{a^2}(ax-1)$
$\int\ln{ax}\,dx=x\ln{ax}-x$
$\int\frac{dx}{a^2+x^2}=\frac{1}{a}\tan^{-1}\frac{x}{a}$
$\int\frac{dx}{a^2-x^2}-\frac{1}{2a}\ln\left|\frac{x+a}{x-a}\right|$
$\int\frac{dx}{\sqrt{a^2+x^2}}=\sinh^{-1}\frac{x}{a}$
$\int\frac{dx}{\sqrt{a^2-x^2}}=\sin^{-1}\frac{x}{a}$
$\int\sqrt{a^2+x^2}\,dx=\frac{x}{2}\sqrt{a^2+x^2}+\frac{a^2}{2}\sinh^{-1}\frac{x}{a}$
$\int\sqrt{a^2-x^2}\,dx=\frac{x}{2}\int{a^2-x^2}+\frac{a^2}{2}\sin^{-1}\frac{x}{a}$

Candela Citations

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Introduction to Electricity, Magnetism, and Circuits by Daryl Janzen is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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