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. $\text{sin}({90}^0-\theta )=\text{cos}\theta$
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. $\text{sin}(\alpha \pm \beta )=\text{sin}\alpha \text{cos}\beta \pm \text{cos}\alpha \text{sin}\beta$
11. $\text{cos}(\alpha \pm \beta )=\text{cos}\alpha \text{cos}\beta \mp \text{sin}\alpha \text{sin}\beta$
12. $\text{tan}(\alpha \pm \beta )=\frac{\text{tan}\alpha \pm \text{tan}\beta }{1\mp \text{tan}\alpha \text{tan}\beta }$
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. $\text{sin}\alpha +\text{sin}\beta =2\text{sin}\frac{1}{2}(\alpha +\beta )\text{cos}\frac{1}{2}(\alpha -\beta )$
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: $c^2=a^2+b^2-2ab\text{cos}\gamma$
3. Pythagorean theorem: $a^2+b^2=c^2$

Series expansions

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

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. $\int \text{sin}xdx=\text{−}\text{cos}x$
5. $\int \text{cos}xdx=\text{sin}x$
6. $\int \text{tan}xdx=\text{ln}|\text{sec}x|$
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. $\int x{e}^{ax}dx=\frac{e^{ax}}{a^2}(ax-1)$
12. $\int \text{ln}axdx=x\text{ln}ax-x$
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. $\int \sqrt{a^2-x^2}dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}{\text{sin}}^{-1}\frac{x}{a}$