Mathematical Formulas

Quadratic formula

If [latex]a{x}^{2}+bx+c=0,[/latex] then [latex]x=\frac{\text{−}b±\sqrt{{b}^{2}-4ac}}{2a}[/latex]

Trigonometry

Trigonometric Identities

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

Triangles

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

Series expansions

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

Derivatives

1. [latex]\frac{d}{dx}[af(x)]=a\frac{d}{dx}f(x)[/latex]
2. [latex]\frac{d}{dx}[f(x)+g(x)]=\frac{d}{dx}f(x)+\frac{d}{dx}g(x)[/latex]
3. [latex]\frac{d}{dx}[f(x)g(x)]=f(x)\frac{d}{dx}g(x)+g(x)\frac{d}{dx}f(x)[/latex]
4. [latex]\frac{d}{dx}f(u)=[\frac{d}{du}f(u)]\frac{du}{dx}[/latex]
5. [latex]\frac{d}{dx}{x}^{m}=m{x}^{m-1}[/latex]
6. [latex]\frac{d}{dx}\,\text{sin}\,x=\text{cos}\,x[/latex]
7. [latex]\frac{d}{dx}\,\text{cos}\,x=\text{−}\text{sin}\,x[/latex]
8. [latex]\frac{d}{dx}\,\text{tan}\,x={\text{sec}}^{2}\,x[/latex]
9. [latex]\frac{d}{dx}\,\text{cot}\,x=\text{−}{\text{csc}}^{2}\,x[/latex]
10. [latex]\frac{d}{dx}\,\text{sec}\,x=\text{tan}\,x\,\text{sec}\,x[/latex]
11. [latex]\frac{d}{dx}\,\text{csc}\,x=\text{−}\text{cot}\,x\,\text{csc}\,x[/latex]
12. [latex]\frac{d}{dx}{e}^{x}={e}^{x}[/latex]
13. [latex]\frac{d}{dx}\,\text{ln}\,x=\frac{1}{x}[/latex]
14. [latex]\frac{d}{dx}\,{\text{sin}}^{-1}\,x=\frac{1}{\sqrt{1-{x}^{2}}}[/latex]
15. [latex]\frac{d}{dx}\,{\text{cos}}^{-1}x=-\frac{1}{\sqrt{1-{x}^{2}}}[/latex]
16. [latex]\frac{d}{dx}\,{\text{tan}}^{-1}x=-\frac{1}{1+{x}^{2}}[/latex]

Integrals

1. [latex]\int af(x)dx=a\int f(x)dx[/latex]
2. [latex]\int [f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx[/latex]
3. [latex]\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}[/latex]
4. [latex]\int \text{sin}\,x\,dx=\text{−}\text{cos}\,x[/latex]
5. [latex]\int \text{cos}\,x\,dx=\text{sin}\,x[/latex]
6. [latex]\int \text{tan}\,x\,dx=\text{ln}|\text{sec}\,x|[/latex]
7. [latex]\int {\text{sin}}^{2}\,ax\,dx=\frac{x}{2}-\frac{\text{sin}\,2ax}{4a}[/latex]
8. [latex]\int {\text{cos}}^{2}\,ax\,dx=\frac{x}{2}+\frac{\text{sin}\,2ax}{4a}[/latex]
9. [latex]\int \text{sin}\,ax\,\text{cos}\,ax\,dx=-\frac{\text{cos}2ax}{4a}[/latex]
10. [latex]\int {e}^{ax}\,dx=\frac{1}{a}{e}^{ax}[/latex]
11. [latex]\int x{e}^{ax}dx=\frac{{e}^{ax}}{{a}^{2}}(ax-1)[/latex]
12. [latex]\int \text{ln}\,ax\,dx=x\,\text{ln}\,ax-x[/latex]
13. [latex]\int \frac{dx}{{a}^{2}+{x}^{2}}=\frac{1}{a}\,{\text{tan}}^{-1}\frac{x}{a}[/latex]
14. [latex]\int \frac{dx}{{a}^{2}-{x}^{2}}=\frac{1}{2a}\,\text{ln}|\frac{x+a}{x-a}|[/latex]
15. [latex]\int \frac{dx}{\sqrt{{a}^{2}+{x}^{2}}}={\text{sinh}}^{-1}\frac{x}{a}[/latex]
16. [latex]\int \frac{dx}{\sqrt{{a}^{2}-{x}^{2}}}={\text{sin}}^{-1}\frac{x}{a}[/latex]
17. [latex]\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}[/latex]
18. [latex]\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}[/latex]