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. $\text{sin}\phantom{}\theta =1\text{/}\text{csc}\phantom{}\theta$
2. $\text{cos}\phantom{}\theta =1\text{/}\text{sec}\phantom{}\theta$
3. $\text{tan}\phantom{}\theta =1\text{/}\text{cot}\phantom{}\theta$
4. $\text{sin}({90}^0-\theta )=\text{cos}\phantom{}\theta$
5. $\text{cos}({90}^0-\theta )=\text{sin}\phantom{}\theta$
6. $\text{tan}({90}^0-\theta )=\text{cot}\phantom{}\theta$
7. ${\text{sin}}^2\phantom{}\theta +{\text{cos}}^2\phantom{}\theta =1$
8. ${\text{sec}}^2\phantom{}\theta -{\text{tan}}^2\phantom{}\theta =1$
9. $\text{tan}\phantom{}\theta =\text{sin}\phantom{}\theta \text{/}\text{cos}\phantom{}\theta$
10. $\text{sin}(\alpha \pm \beta )=\text{sin}\phantom{}\alpha \phantom{}\text{cos}\phantom{}\beta \pm \text{cos}\phantom{}\alpha \phantom{}\text{sin}\phantom{}\beta$
11. $\text{cos}(\alpha \pm \beta )=\text{cos}\phantom{}\alpha \phantom{}\text{cos}\phantom{}\beta \mp \text{sin}\phantom{}\alpha \phantom{}\text{sin}\phantom{}\beta$
12. $\text{tan}(\alpha \pm \beta )=\frac{\text{tan}\phantom{}\alpha \pm \text{tan}\phantom{}\beta }{1\mp \text{tan}\phantom{}\alpha \phantom{}\text{tan}\phantom{}\beta }$
13. $\text{sin}\phantom{}2\theta =2\text{sin}\phantom{}\theta \phantom{}\text{cos}\phantom{}\theta$
14. $\text{cos}\phantom{}2\theta ={\text{cos}}^2\phantom{}\theta -{\text{sin}}^2\phantom{}\theta =2\phantom{}{\text{cos}}^2\phantom{}\theta -1=1-2\phantom{}{\text{sin}}^2\phantom{}\theta$
15. $\text{sin}\phantom{}\alpha +\text{sin}\phantom{}\beta =2\phantom{}\text{sin}\frac{1}{2}(\alpha +\beta )\text{cos}\frac{1}{2}(\alpha -\beta )$
16. $\text{cos}\phantom{}\alpha +\text{cos}\phantom{}\beta =2\phantom{}\text{cos}\frac{1}{2}(\alpha +\beta )\text{cos}\frac{1}{2}(\alpha -\beta )$

Triangles

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

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

Derivatives

1. $\frac{d}{dx}\left[af\left(x\right)\right]=a\frac{d}{dx}f\left(x\right)$
2. $\frac{d}{dx}[f\left(x\right)+g\left(x\right)]=\frac{d}{dx}f\left(x\right)+\frac{d}{dx}g\left(x\right)$
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. $\frac{d}{dx}\phantom{}\text{sin}\phantom{}x=\text{cos}\phantom{}x$
7. $\frac{d}{dx}\phantom{}\text{cos}\phantom{}x=\text{−}\text{sin}\phantom{}x$
8. $\frac{d}{dx}\phantom{}\text{tan}\phantom{}x={\text{sec}}^2\phantom{}x$
9. $\frac{d}{dx}\phantom{}\text{cot}\phantom{}x=\text{−}{\text{csc}}^2\phantom{}x$
10. $\frac{d}{dx}\phantom{}\text{sec}\phantom{}x=\text{tan}\phantom{}x\phantom{}\text{sec}\phantom{}x$
11. $\frac{d}{dx}\phantom{}\text{csc}\phantom{}x=\text{−}\text{cot}\phantom{}x\phantom{}\text{csc}\phantom{}x$
12. $\frac{d}{dx}{e}^x=e^x$
13. $\frac{d}{dx}\phantom{}\text{ln}\phantom{}x=\frac{1}{x}$
14. $\frac{d}{dx}\phantom{}{\text{sin}}^{-1}\phantom{}x=\frac{1}{\sqrt{1-x^2}}$
15. $\frac{d}{dx}\phantom{}{\text{cos}}^{-1}x=-\frac{1}{\sqrt{1-x^2}}$
16. $\frac{d}{dx}\phantom{}{\text{tan}}^{-1}x=-\frac{1}{1+x^2}$

Integrals

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