Appendix E: Mathematical Formulas

Quadratic formula

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

This table has three columns and four rows. The entry in the first row are: Triangle of base b and height h, Area equal to half bh. The third cell in the first row is blank. the second row has the following entries: Circle of radius r, Circumference equal to 2 pi r, Area equal to pi r squared. The third row has the following entries: Sphere of radius r, Surface area equal to 4 pi r squared, volume equal to 4 by 3 pi r cubed. The fourth row has the following entries: Cylinder of radius r and height h, Area of curved surface equal to 2 pi r h, Volume equal to pi r squared 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. $\text{sin}\left({90}^0-\theta \right)=\text{cos}\theta$
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: $c^2=a^2+b^2-2ab\text{cos}\gamma$
   
3. Pythagorean theorem: $a^2+b^2=c^2$
   

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