Mathematical Formulas

Quadratic formula

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

    Figure shows a triangle with three dissimilar sides labeled a, b and c. All three angles of the triangle are acute angles. The angle between b and c is alpha, the angle between a and c is beta and the angle between a and b is gamma.

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

    Figure shows a right triangle. Its three sides are labeled a, b and c with c being the hypotenuse. The angle between a and c is labeled theta.

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. $${(1±x)}^{n}=1±\frac{nx}{1\text{!}}+\frac{n(n-1){x}^{2}}{2\text{!}}±\text{···}({x}^{2}<1)$$
  3. $${(1±x)}^{\text{−}n}=1\mp \frac{nx}{1\text{!}}+\frac{n(n+1){x}^{2}}{2\text{!}}\mp \text{···}({x}^{2}<1)$$
  4. $$\text{sin}\,x=x-\frac{{x}^{3}}{3\text{!}}+\frac{{x}^{5}}{5\text{!}}-\text{···}$$
  5. $$\text{cos}\,x=1-\frac{{x}^{2}}{2\text{!}}+\frac{{x}^{4}}{4\text{!}}-\text{···}$$
  6. $$\text{tan}\,x=x+\frac{{x}^{3}}{3}+\frac{2{x}^{5}}{15}+\text{···}$$
  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. $$\frac{d}{dx}[af(x)]=a\frac{d}{dx}f(x)$$
  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. $$\int \sqrt{{a}^{2}-{x}^{2}}\,dx=\frac{x}{2}\sqrt{{a}^{2}-{x}^{2}}+\frac{{a}^{2}}{2}\,{\text{sin}}^{-1}\frac{x}{a}$$