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E – Mathematical Formulas

Quadratic formula

If ax^2+bx+c=0, then x=\pm\frac{-b\pm\sqrt{b^2-4ac}}{2a}

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

  1. \sin\theta=1/\csc\theta
  2. \cos\theta=1/\sec\theta
  3. \tan\theta=1/\cot\theta
  4. \sin(90^{\circ}-\theta)=\cos\theta
  5. \cos(90^{\circ}-\theta)=\sin\theta
  6. \tan(90^{\circ}-\theta)=\cot\theta
  7. \sin^2\theta+\cos^2\theta=1
  8. \sec^2\theta-\tan^2\theta=1
  9. \tan\theta=\sin\theta/\cos\theta
  10. \sin(\alpha\pm\beta)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta
  11. \cos(\alpha\pm\beta)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta
  12. \tan(\alpha\pm\beta)=\frac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\tan\beta}
  13. \sin2\theta=2\sin\theta\cos\theta
  14. \cos2\theta=\cos^2\theta-\sin^2\theta=2\cos^2\theta-1=1-2\sin^2\theta
  15. \sin\alpha+\sin\beta=2\sin\frac{1}{2}(\alpha+\beta)\cos\frac{1}{2}(\alpha-\beta)
  16. \cos\alpha+\cos\beta=2\cos\frac{1}{2}(\alpha+\beta)\cos\frac{1}{2}(\alpha-\beta)

Triangles

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

    Figure shows a triangle with three dissimilar sides labeled a, b and c. All three angles of the triangle are acute angles. The angle between b and c is alpha, the angle between a and c is beta and the angle between a and b is gamma.

  3. Pythagorean theorem: a^2+b^2=c^2

    Figure shows a right triangle. Its three sides are labeled a, b and c with c being the hypotenuse. The angle between a and c is labeled theta.

Series expansions

  1. 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
  2. (1\pm x)^n=1\pm\frac{nx}{1!}+\frac{n(n-1)x^2}{2!}\pm\ldots\ (x^2<1)
  3. (1\pm x)^{-n}=1\mp\frac{nx}{1!}+\frac{n(n-1)x^2}{2!}\mp\ldots\ (x^2<1)
  4. \sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\ldots
  5. \cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\ldots
  6. \tan x=x+\frac{x^3}{3}+\frac{2x^5}{15}+\ldots
  7. e^x=1+x+\frac{x^2}{2!}+\ldots
  8. \ln(1+x)=x-\frac{1}{2}x^2+\frac{1}{3}x^3-\ldots\ (|x|<1)

Derivatives

  1. \frac{d}{dx}[af(x)]=a\frac{d}{dx}f(x)
  2. \frac{d}{dx}[f(x)+g(x)]=\frac{d}{dx}f(x)+\frac{d}{dx}g(x)
  3. \frac{d}{dx}[f(x)g(x)]=f(x)\frac{d}{dx}g(x)+g(x)\frac{d}{dx}f(x)
  4. \frac{d}{dx}f(u)=\left[\frac{d}{du}f(u)\right]\frac{du}{dx}
  5. \frac{d}{dx}x^m=mx^{m-1}
  6. \frac{d}{dx}\sin x=\cos x
  7. \frac{d}{dx}\cos x=-\sin x
  8. \frac{d}{dx}\tan x=\sec^2x
  9. \frac{d}{dx}\cot x=-\csc^2x
  10. \frac{d}{dx}\sec x=\tan x\sec x
  11. \frac{d}{dx}\csc x=-\cot x\csc x
  12. \frac{d}{dx}e^x=e^x
  13. \frac{d}{dx}\ln x=\frac{1}{x}
  14. \frac{d}{dx}\sin^{-1}x=\frac{1}{\sqrt{1-x^2}}
  15. \frac{d}{dx}\cos^{-1}x=-\frac{1}{\sqrt{1-x^2}}
  16. \frac{d}{dx}\tan^{-1}x=-\frac{1}{1+x^2}

Integrals

  1. \int af(x)\,dx=a\int f(x)\,dx
  2. \int[f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx
  3. \int x^m\,dx=\frac{x^{m+1}}{m+1}\ (m\neq1)
    ~~~~~~=\ln x\ (m=-1)
  4. \int\sin{x}\,dx=-\cos x
  5. \int\cos{x}\,dx=\sin x
  6. \int\tan{x}\,dx=\ln|\sec x|
  7. \int\sin^2{ax}\,dx=\frac{x}{2}-\frac{\sin{2ax}}{4a}
  8. \int\cos^2{ax}\,dx=\frac{x}{2}+\frac{\sin{2ax}}{4a}
  9. \int\sin{ax}\cos{ax}\,dx=-\frac{\cos2ax}{4a}
  10. \int e^{ax}\,dx=\frac{1}{a}e^{ax}
  11. \int xe^{ax}\,dx=\frac{e^{ax}}{a^2}(ax-1)
  12. \int\ln{ax}\,dx=x\ln{ax}-x
  13. \int\frac{dx}{a^2+x^2}=\frac{1}{a}\tan^{-1}\frac{x}{a}
  14. \int\frac{dx}{a^2-x^2}-\frac{1}{2a}\ln\left|\frac{x+a}{x-a}\right|
  15. \int\frac{dx}{\sqrt{a^2+x^2}}=\sinh^{-1}\frac{x}{a}
  16. \int\frac{dx}{\sqrt{a^2-x^2}}=\sin^{-1}\frac{x}{a}
  17. \int\sqrt{a^2+x^2}\,dx=\frac{x}{2}\sqrt{a^2+x^2}+\frac{a^2}{2}\sinh^{-1}\frac{x}{a}
  18. \int\sqrt{a^2-x^2}\,dx=\frac{x}{2}\int{a^2-x^2}+\frac{a^2}{2}\sin^{-1}\frac{x}{a}

License

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Introduction to Electricity, Magnetism, and Circuits Copyright © 2018 by Daryl Janzen is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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