Mathematical Formulas

Quadratic formula

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

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. $\text{sin}\left({90}^0-\theta \right)=\text{cos}\theta$
5. $\text{cos}\left({90}^0-\theta \right)=\text{sin}\theta$
6. $\text{tan}\left({90}^0-\theta \right)=\text{cot}\theta$
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. $\text{sin}\left(\alpha \pm \beta \right)=\text{sin}\alpha \text{cos}\beta \pm \text{cos}\alpha \text{sin}\beta$
11. $\text{cos}\left(\alpha \pm \beta \right)=\text{cos}\alpha \text{cos}\beta \mp \text{sin}\alpha \text{sin}\beta$
12. $\text{tan}\left(\alpha \pm \beta \right)=\frac{\text{tan}\alpha \pm \text{tan}\beta }{1\mp \text{tan}\alpha \text{tan}\beta }$
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. $\text{sin}\alpha +\text{sin}\beta =2\text{sin}\frac{1}{2}\left(\alpha +\beta \right)\text{cos}\frac{1}{2}\left(\alpha -\beta \right)$
16. $\text{cos}\alpha +\text{cos}\beta =2\text{cos}\frac{1}{2}\left(\alpha +\beta \right)\text{cos}\frac{1}{2}\left(\alpha -\beta \right)$
17. $\text{s}=r\theta$

Triangles

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

   View original figure (external; image not reproduced in this text edition)

   Original figure description: 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: $a^2+b^2=c^2$
   

   View original figure (external; image not reproduced in this text edition)

   Original figure description: 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: ${\left(a+b\right)}^n=a^n+n{a}^{n-1}b+\frac{n\left(n-1\right)a^{n-2}{b}^2}{2\text{!}}+\frac{n\left(n-1\right)\left(n-2\right)a^{n-3}{b}^3}{3\text{!}}+\text{···}$
2. ${\left(1\pm x\right)}^n=1\pm \frac{nx}{1\text{!}}+\frac{n\left(n-1\right)x^2}{2\text{!}}\pm \text{···}\left(x^2<1\right)$
3. ${\left(1\pm x\right)}^{\text{−}n}=1\mp \frac{nx}{1\text{!}}+\frac{n\left(n+1\right)x^2}{2\text{!}}\mp \text{···}\left(x^2<1\right)$
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. $\text{tan}x=x+\frac{x^3}{3}+\frac{2x^5}{15}+\text{···}$
7. $e^x=1+x+\frac{x^2}{2\text{!}}+\text{···}$
8. $\text{ln}\left(1+x\right)=x-\frac{1}{2}{x}^2+\frac{1}{3}{x}^3-\text{···}\left(\left|x\right|<1\right)$

Derivatives

1. $\frac{d}{dx}\left[af\left(x\right)\right]=a\frac{d}{dx}f\left(x\right)$
2. $\frac{d}{dx}\left[f\left(x\right)+g\left(x\right)\right]=\frac{d}{dx}f\left(x\right)+\frac{d}{dx}g\left(x\right)$
3. $\frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]=f\left(x\right)\frac{d}{dx}g\left(x\right)+g\left(x\right)\frac{d}{dx}f\left(x\right)$
4. $\frac{d}{dx}f\left(u\right)=\left[\frac{d}{du}f\left(u\right)\right]\frac{du}{dx}$
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. $\frac{d}{dx}{\text{sin}}^{\mathrm{-1}}x=\frac{1}{\sqrt{1-x^2}}$
15. $\frac{d}{dx}{\text{cos}}^{\mathrm{-1}}x=-\frac{1}{\sqrt{1-x^2}}$
16. $\frac{d}{dx}{\text{tan}}^{\mathrm{-1}}x=\frac{1}{1+x^2}$

Integrals

1. $\int af\left(x\right)dx=a\int f\left(x\right)dx$
2. $\int \left[f\left(x\right)+g\left(x\right)\right]dx=\int f\left(x\right)dx+\int g\left(x\right)dx$
3. $\begin{array}{cc}\int x^{m}dx& =\frac{x^{m+1}}{m+1}\left(m\ne \text{−}1\right)\\ & =\text{ln}x(m=\mathrm{-1})\end{array}$
4. $\int \text{sin}xdx=\text{−}\text{cos}x$
5. $\int \text{cos}xdx=\text{sin}x$
6. $\int \text{tan}xdx=\text{ln}\left|\text{sec}x\right|$
7. $\int {\text{sin}}^{2}axdx=\frac{x}{2}-\frac{\text{sin}2ax}{4a}$
8. $\int {\text{cos}}^{2}axdx=\frac{x}{2}+\frac{\text{sin}2ax}{4a}$
9. $\int \text{sin}ax\text{cos}axdx=-\frac{\text{cos}2ax}{4a}$
10. $\int e^{ax}dx=\frac{1}{a}{e}^{ax}$
11. $\int x{e}^{ax}dx=\frac{e^{ax}}{a^2}\left(ax-1\right)$
12. $\int \text{ln}axdx=x\text{ln}ax-x$
13. $\int \frac{dx}{a^2+x^2}=\frac{1}{a}{\text{tan}}^{\mathrm{-1}}\frac{x}{a}$
14. $\int \frac{dx}{a^2-x^2}=\frac{1}{2a}\text{ln}\left|\frac{x+a}{x-a}\right|$
15. $\int \frac{dx}{\sqrt{a^2+x^2}}={\text{sinh}}^{\mathrm{-1}}\frac{x}{a}$
16. $\int \frac{dx}{\sqrt{a^2-x^2}}={\text{sin}}^{\mathrm{-1}}\frac{x}{a}$
17. $\int \sqrt{a^2+x^2}dx=\frac{x}{2}\sqrt{a^2+x^2}+\frac{a^2}{2}{\text{sinh}}^{\mathrm{-1}}\frac{x}{a}$
18. $\int \sqrt{a^2-x^2}dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}{\text{sin}}^{\mathrm{-1}}\frac{x}{a}$
19. $\int \frac{1}{(x^2+a^2{)}^{3/2}}dx=\frac{x}{a^2\sqrt{x^2+a^2}}$
20. $\int \frac{x}{(x^2+a^2{)}^{3/2}}dx=–\frac{1}{\sqrt{x^2+a^2}}$