Mathematical Formulas

Quadratic formula

If

    \[a{x}^{2}+bx+c=0,\]

then

    \[x=\frac{\text{−}b±\sqrt{{b}^{2}-4ac}}{2a}\]

Geometry
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

    \[=\pi {r}^{2}h\]

Trigonometry

Trigonometric Identities

  1.     \[\text{sin}\,\theta =1\text{/}\text{csc}\,\theta\]

  2.     \[\text{cos}\,\theta =1\text{/}\text{sec}\,\theta\]

  3.     \[\text{tan}\,\theta =1\text{/}\text{cot}\,\theta\]

  4.     \[\text{sin}({90}^{0}-\theta )=\text{cos}\,\theta\]

  5.     \[\text{cos}({90}^{0}-\theta )=\text{sin}\,\theta\]

  6.     \[\text{tan}({90}^{0}-\theta )=\text{cot}\,\theta\]

  7.     \[{\text{sin}}^{2}\,\theta +{\text{cos}}^{2}\,\theta =1\]

  8.     \[{\text{sec}}^{2}\,\theta -{\text{tan}}^{2}\,\theta =1\]

  9.     \[\text{tan}\,\theta =\text{sin}\,\theta \text{/}\text{cos}\,\theta\]

  10.     \[\text{sin}(\alpha ±\beta )=\text{sin}\,\alpha \,\text{cos}\,\beta ±\text{cos}\,\alpha \,\text{sin}\,\beta\]

  11.     \[\text{cos}(\alpha ±\beta )=\text{cos}\,\alpha \,\text{cos}\,\beta \mp \text{sin}\,\alpha \,\text{sin}\,\beta\]

  12.     \[\text{tan}(\alpha ±\beta )=\frac{\text{tan}\,\alpha ±\text{tan}\,\beta }{1\mp \text{tan}\,\alpha \,\text{tan}\,\beta }\]

  13.     \[\text{sin}\,2\theta =2\text{sin}\,\theta \,\text{cos}\,\theta\]

  14.     \[\text{cos}\,2\theta ={\text{cos}}^{2}\,\theta -{\text{sin}}^{2}\,\theta =2\,{\text{cos}}^{2}\,\theta -1=1-2\,{\text{sin}}^{2}\,\theta\]

  15.     \[\text{sin}\,\alpha +\text{sin}\,\beta =2\,\text{sin}\frac{1}{2}(\alpha +\beta )\text{cos}\frac{1}{2}(\alpha -\beta )\]

  16.     \[\text{cos}\,\alpha +\text{cos}\,\beta =2\,\text{cos}\frac{1}{2}(\alpha +\beta )\text{cos}\frac{1}{2}(\alpha -\beta )\]

Triangles

  1. Law of sines:

        \[\frac{a}{\text{sin}\,\alpha }=\frac{b}{\text{sin}\,\beta }=\frac{c}{\text{sin}\,\gamma }\]

  2. Law of cosines:

        \[{c}^{2}={a}^{2}+{b}^{2}-2ab\,\text{cos}\,\gamma\]

    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:

        \[{(a+b)}^{n}={a}^{n}+n{a}^{n-1}b+\frac{n(n-1){a}^{n-2}{b}^{2}}{2\text{!}}+\frac{n(n-1)(n-2){a}^{n-3}{b}^{3}}{3\text{!}}+\text{···}\]

  2.     \[{(1±x)}^{n}=1±\frac{nx}{1\text{!}}+\frac{n(n-1){x}^{2}}{2\text{!}}±\text{···}({x}^{2}<1)\]

  3.     \[{(1±x)}^{\text{−}n}=1\mp \frac{nx}{1\text{!}}+\frac{n(n+1){x}^{2}}{2\text{!}}\mp \text{···}({x}^{2}<1)\]

  4.     \[\text{sin}\,x=x-\frac{{x}^{3}}{3\text{!}}+\frac{{x}^{5}}{5\text{!}}-\text{···}\]

  5.     \[\text{cos}\,x=1-\frac{{x}^{2}}{2\text{!}}+\frac{{x}^{4}}{4\text{!}}-\text{···}\]

  6.     \[\text{tan}\,x=x+\frac{{x}^{3}}{3}+\frac{2{x}^{5}}{15}+\text{···}\]

  7.     \[{e}^{x}=1+x+\frac{{x}^{2}}{2\text{!}}+\text{···}\]

  8.     \[\text{ln}(1+x)=x-\frac{1}{2}{x}^{2}+\frac{1}{3}{x}^{3}-\text{···}(|x|<1)\]

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.     \[\frac{d}{dx}f(u)=[\frac{d}{du}f(u)]\frac{du}{dx}\]

  5.     \[\frac{d}{dx}{x}^{m}=m{x}^{m-1}\]

  6.     \[\frac{d}{dx}\,\text{sin}\,x=\text{cos}\,x\]

  7.     \[\frac{d}{dx}\,\text{cos}\,x=\text{−}\text{sin}\,x\]

  8.     \[\frac{d}{dx}\,\text{tan}\,x={\text{sec}}^{2}\,x\]

  9.     \[\frac{d}{dx}\,\text{cot}\,x=\text{−}{\text{csc}}^{2}\,x\]

  10.     \[\frac{d}{dx}\,\text{sec}\,x=\text{tan}\,x\,\text{sec}\,x\]

  11.     \[\frac{d}{dx}\,\text{csc}\,x=\text{−}\text{cot}\,x\,\text{csc}\,x\]

  12.     \[\frac{d}{dx}{e}^{x}={e}^{x}\]

  13.     \[\frac{d}{dx}\,\text{ln}\,x=\frac{1}{x}\]

  14.     \[\frac{d}{dx}\,{\text{sin}}^{-1}\,x=\frac{1}{\sqrt{1-{x}^{2}}}\]

  15.     \[\frac{d}{dx}\,{\text{cos}}^{-1}x=-\frac{1}{\sqrt{1-{x}^{2}}}\]

  16.     \[\frac{d}{dx}\,{\text{tan}}^{-1}x=-\frac{1}{1+{x}^{2}}\]

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.     \[\begin{array}{cc}\hfill \int {x}^{m}dx& =\frac{{x}^{m+1}}{m+1}\,(m\ne \text{−}1)\hfill \\ & =\text{ln}\,x(m=-1)\hfill \end{array}\]

  4.     \[\int \text{sin}\,x\,dx=\text{−}\text{cos}\,x\]

  5.     \[\int \text{cos}\,x\,dx=\text{sin}\,x\]

  6.     \[\int \text{tan}\,x\,dx=\text{ln}|\text{sec}\,x|\]

  7.     \[\int {\text{sin}}^{2}\,ax\,dx=\frac{x}{2}-\frac{\text{sin}\,2ax}{4a}\]

  8.     \[\int {\text{cos}}^{2}\,ax\,dx=\frac{x}{2}+\frac{\text{sin}\,2ax}{4a}\]

  9.     \[\int \text{sin}\,ax\,\text{cos}\,ax\,dx=-\frac{\text{cos}2ax}{4a}\]

  10.     \[\int {e}^{ax}\,dx=\frac{1}{a}{e}^{ax}\]

  11.     \[\int x{e}^{ax}dx=\frac{{e}^{ax}}{{a}^{2}}(ax-1)\]

  12.     \[\int \text{ln}\,ax\,dx=x\,\text{ln}\,ax-x\]

  13.     \[\int \frac{dx}{{a}^{2}+{x}^{2}}=\frac{1}{a}\,{\text{tan}}^{-1}\frac{x}{a}\]

  14.     \[\int \frac{dx}{{a}^{2}-{x}^{2}}=\frac{1}{2a}\,\text{ln}|\frac{x+a}{x-a}|\]

  15.     \[\int \frac{dx}{\sqrt{{a}^{2}+{x}^{2}}}={\text{sinh}}^{-1}\frac{x}{a}\]

  16.     \[\int \frac{dx}{\sqrt{{a}^{2}-{x}^{2}}}={\text{sin}}^{-1}\frac{x}{a}\]

  17.     \[\int \sqrt{{a}^{2}+{x}^{2}}\,dx=\frac{x}{2}\sqrt{{a}^{2}+{x}^{2}}+\frac{{a}^{2}}{2}\,{\text{sinh}}^{-1}\frac{x}{a}\]

  18.     \[\int \sqrt{{a}^{2}-{x}^{2}}\,dx=\frac{x}{2}\sqrt{{a}^{2}-{x}^{2}}+\frac{{a}^{2}}{2}\,{\text{sin}}^{-1}\frac{x}{a}\]

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University Physics Volume 1 by OpenStax is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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