# Mathematical Formulas

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

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

Series expansions

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

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