Approximations

Irrational

Ratios and Angles

Basic Formulae

$\begin{array}{rl}{\sin }^2\theta +{\cos }^2\theta & =1\\ {\tan }^2\theta -{\sec }^2\theta & =1\\ {\csc }^2\theta -{\cot }^2\theta & =1\end{array}$

$\begin{array}{rlr}\sin x& =\cos ({90}^{\circ }-x)& ={\displaystyle \frac{1}{\csc x}}\\ \cos x& =\sin ({90}^{\circ }-x)& ={\displaystyle \frac{1}{\sec x}}\\ \tan x& =\cot ({90}^{\circ }-x)& ={\displaystyle \frac{\sin x}{\cos x}}={\displaystyle \frac{1}{\cot x}}\end{array}$

A system of rectangular coordinate axes divides a plane into four quadrants. An angle $\theta$ lies in one and only one of these quadrants.
The change and sign of the trigonometric ratios in the various quadrants are shown in Fig-1 below.

Sum and Difference

Two Angle

$\begin{array}{rl}\sin (A\pm B)& =\sin A\cos B\pm \cos A\sin B\\ \cos (A\pm B)& =\cos A\cos B\mp \sin A\sin B\end{array}$

$\begin{array}{rl}\tan (A\pm B)& ={\displaystyle \frac{\tan A\pm \tan B}{1\mp \tan A\tan B}}\\ \cot (A\pm B)& ={\displaystyle \frac{\cot A\cot B\mp 1}{\cot B\pm \cot A}}\end{array}$

$\begin{array}{rlrlr}\sin (A+B)\sin (A-B)& & ={\sin }^{2}A-{\sin }^{2}B& & ={\cos }^{2}B-{\cos }^{2}A\\ \cos (A+B)\cos (A-B)& & ={\cos }^{2}A-{\sin }^{2}B& & ={\cos }^{2}B-{\sin }^{2}A\end{array}$

Three Angle

$\begin{array}{rl}\sin (A+B+C)& =\sin A\cos B\cos C+\cos A\sin B\cos C+\cos A\cos B\sin C-\sin A\sin B\sin C\\ & =\cos A\cos B\cos C(\tan A+\tan B+\tan C-\tan A\tan B\tan C)\end{array}$

$\begin{array}{rl}\cos (A+B+C)& =\cos A\cos B\cos C-\sin A\sin B\cos C-\sin A\cos B\sin C-\cos A\sin B\sin C\\ & =\cos A\cos B\cos C(1-\tan A\tan B-\tan B\tan C-\tan C\tan A)\end{array}$

$\begin{array}{rl}\tan (A+B+C)& ={\displaystyle \frac{\tan A+\tan B+\tan C-\tan A\tan B\tan C}{1-\tan A\tan B-\tan B\tan C-\tan C\tan A}}\\ \cot (A+B+C)& ={\displaystyle \frac{\cot A+\cot B+\cot C-\cot A\cot B\cot C}{1-\cot A\cot B-\cot B\cot C-\cot C\cot A}}\end{array}$

General

$\begin{array}{rl}{S}_0& =1\\ S_1& =\sum _i{x}_i& & =\sum _i\tan {\theta }_i\\ S_2& =\sum _{i<j}{x}_i{x}_j& & =\sum _{i<j}\tan {\theta }_i\tan {\theta }_j\\ S_3& =\sum _{i<j<k}{x}_i{x}_j{x}_k& & =\sum _{i<j<k}\tan {\theta }_i\tan {\theta }_j\tan {\theta }_k\\ & ⋮& & ⋮\end{array}$

Then,

$\begin{array}{rl}\tan ({\theta }_1+{\theta }_2+\cdots +{\theta }_n)& ={\displaystyle \frac{S_1-S_3+S_5-S_7+\cdots }{1-S_2+S_4-S_6+\cdots }}\\ \sec ({\theta }_1+{\theta }_2+\cdots +{\theta }_n)& ={\displaystyle \frac{\sec {\theta }_1\sec {\theta }_2\cdots \sec {\theta }_n}{S_0-S_2+S_4-\cdots }}\\ \csc ({\theta }_1+{\theta }_2+\cdots +{\theta }_n)& ={\displaystyle \frac{\sec {\theta }_1\sec {\theta }_2\cdots \sec {\theta }_n}{S_1-S_3+S_5-\cdots }}\end{array}$

$\cos 2^r\theta \cdot \cos 2^{r+1}\theta \cdot \cos 2^{r+2}\theta \cdots \cos 2^n\theta ={\displaystyle \frac{\sin 2^{n+1}\theta }{2^{n-r+1}\sin 2^r\theta }}$

$\begin{array}{rl}\sin \left(\alpha \right)+\sin (\alpha +\beta )+\sin (\alpha +2\beta )+\cdots +\sin (\alpha +n\beta )& ={\displaystyle \frac{\sin ((n+1)\frac{\beta }{2})}{\sin \left(\frac{\beta }{2}\right)}}\sin (\alpha +n\frac{\beta }{2})\\ \cos \left(\alpha \right)+\cos (\alpha +\beta )+\cos (\alpha +2\beta )+\cdots +\cos (\alpha +n\beta )& ={\displaystyle \frac{\sin ((n+1)\frac{\beta }{2})}{\sin \left(\frac{\beta }{2}\right)}}\cos (\alpha +n\frac{\beta }{2})\end{array}$

$1+2\cos \theta +2\cos \left(2\theta \right)+2\cos \left(3\theta \right)+\cdots +2\cos \left(n\theta \right)={\displaystyle \frac{\sin \left((n+\frac{1}{2})\theta \right)}{\sin \left(\frac{\theta }{2}\right)}}$

Generally not true! only for common values of ‘$\theta$’. USE WITH CAUTION

${\displaystyle \frac{\sin {\theta }_1\pm \sin {\theta }_2+\sin {\theta }_3+\cdots +\sin {\theta }_n}{\cos {\theta }_1\pm \cos {\theta }_2+\cos {\theta }_3+\cdots +\cos {\theta }_n}}=\tan {\displaystyle \frac{{\theta }_1\pm {\theta }_2+{\theta }_3+\cdots +{\theta }_n}{n}}$

Product to Sum

$\begin{array}{rl}2\sin A\cos B& =\sin (A+B)+\sin (A-B)\\ 2\cos A\sin B& =\sin (A+B)-\sin (A-B)\\ 2\cos A\cos B& =\cos (A+B)+\cos (A-B)\\ 2\sin A\sin B& =\cos (A-B)-\cos (A+B)\end{array}$

Sum to Product

$\begin{array}{rl}\sin C+\sin D& =2\sin \frac{C+D}{2}\cos \frac{C-D}{2}\\ \sin C-\sin D& =2\cos \frac{C+D}{2}\sin \frac{C-D}{2}\\ \cos C+\cos D& =2\cos \frac{C+D}{2}\cos \frac{C-D}{2}\\ \cos C-\cos D& =-2\sin \frac{C+D}{2}\sin \frac{C-D}{2}\end{array}$

Multiple Angle

Two/Three Angle

$\begin{array}{rl}\sin 2\theta & =2\sin \theta \cos \theta \\ & ={\displaystyle \frac{2\tan \theta }{1+{\tan }^2\theta }}\end{array}$

$\begin{array}{rl}\cos 2\theta & ={\cos }^2\theta -{\sin }^2\theta \\ & =2{\cos }^2\theta -1\\ & =1-2{\sin }^2\theta \\ & ={\displaystyle \frac{1-{\tan }^2\theta }{1+{\tan }^2\theta }}\end{array}$

$\tan 2\theta ={\displaystyle \frac{2\tan \theta }{1-{\tan }^2\theta }}$

$\begin{array}{rl}\sin 3\theta & =3\sin \theta -4{\sin }^3\theta \\ & =4\sin ({60}^{\circ }-\theta )\sin \left(\theta \right)\sin ({60}^{\circ }+\theta )\end{array}$

$\begin{array}{rl}\cos 3\theta & =4{\cos }^3\theta -3\cos \theta \\ & =4\cos ({60}^{\circ }-\theta )\cos \left(\theta \right)\cos ({60}^{\circ }+\theta )\end{array}$

$\begin{array}{rl}\tan 3\theta & ={\displaystyle \frac{3\tan \theta -{\tan }^3\theta }{1-3{\tan }^2\theta }}\\ & =\tan ({60}^{\circ }-\theta )\tan \left(\theta \right)\tan ({60}^{\circ }+\theta )\end{array}$

Multi(>3) Angle

$\begin{array}{rl}\sin 4\theta & =4\sin \theta {\cos }^3\theta -4\cos \theta {\sin }^3\theta \\ \cos 4\theta & =8{\cos }^4\theta -8{\cos }^2\theta +1\\ \tan 4\theta & ={\displaystyle \frac{4\tan \theta -4{\tan }^3\theta }{1-6{\tan }^2\theta +{\tan }^4\theta }}\\ \sin 5\theta & =16{\sin }^5\theta -20{\sin }^3\theta +5\sin \theta \\ \cos 5\theta & =16{\cos }^5\theta -20{\cos }^3\theta +5\cos \theta \end{array}$

Half Angle

$\begin{array}{rl}\sin {\displaystyle \frac{\theta }{2}}& =\mathrm{sgn}(\sin {\displaystyle \frac{\theta }{2}})\sqrt{{\displaystyle \frac{1-\cos \theta }{2}}}\\ \cos {\displaystyle \frac{\theta }{2}}& =\mathrm{sgn}(\cos {\displaystyle \frac{\theta }{2}})\sqrt{{\displaystyle \frac{1+\cos \theta }{2}}}\end{array}$

$\begin{array}{rl}\tan {\displaystyle \frac{\theta }{2}}& =\frac{1-\cos \theta }{\sin \theta }\\ & =\frac{\sin \theta }{1+\cos \theta }\\ & =\csc \theta -\cot \theta \\ & =\frac{\tan \theta }{1+\sec \theta }\\ & =\mathrm{sgn}(\sin \theta )\sqrt{\frac{1-\cos \theta }{1+\cos \theta }}\end{array}$

Special Cases

$\tan {\displaystyle \frac{\eta \pm \theta }{2}}={\displaystyle \frac{\sin \eta \pm \sin \theta }{\cos \eta +\cos \theta }}$

$\tan ({\displaystyle \frac{\theta }{2}}+{\displaystyle \frac{\pi }{4}})=\sec \theta +\tan \theta$
$\sqrt{{\displaystyle \frac{1-\sin \theta }{1+\sin \theta }}}=\left|{\displaystyle \frac{1-\tan {\displaystyle \frac{\theta }{2}}}{1+\tan {\displaystyle \frac{\theta }{2}}}}\right|$

Power Reduction

Special Cases

If $A+B+C={180}^{\circ }=\pi$ , then

$\begin{array}{rl}\sin 2A+\sin 2B+\sin 2C& =4\sin A\sin B\sin C\\ \sin 2A+\sin 2B-\sin 2C& =4\cos A\cos B\sin C\\ \cos 2A+\cos 2B+\cos 2C& =-1-4\cos A\cos B\cos C\\ \cos 2A+\cos 2B-\cos 2C& =1-4\sin A\sin B\cos C\\ \sin A+\sin B+\sin C& =4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}\\ \sin A+\sin B-\sin C& =4\sin \frac{A}{2}\sin \frac{B}{2}\cos \frac{C}{2}\\ \cos A+\cos B+\cos C& =1+4\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}\\ \cos A+\cos B-\cos C& =-1+4\cos \frac{A}{2}\cos \frac{B}{2}\sin \frac{C}{2}\\ {\sin }^{2}A+{\sin }^{2}B-{\sin }^{2}C& =2\sin A\sin B\cos C\\ {\cos }^{2}A+{\cos }^{2}B-{\cos }^{2}C& =1-2\sin A\sin B\cos C\\ {\cos }^{2}A+{\cos }^{2}B+{\cos }^{2}C& =1-2\cos A\cos B\cos C\\ {\sin }^{2}A+{\sin }^{2}B+{\sin }^{2}C& =2+2\cos A\cos B\cos C\\ \tan A+\tan B+\tan C& =\tan A\tan B\tan C\\ \cot B\cot C+\cot C\cot A+\cot A\cos B& =1\\ \tan \frac{B}{2}\tan \frac{C}{2}+\tan \frac{C}{2}\tan \frac{A}{2}+\tan \frac{A}{2}.\tan \frac{B}{2}& =1\\ \cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}& =\cot \frac{A}{2}\cot \frac{B}{2}\cot \frac{C}{2}\end{array}$

Series Expansion

$\begin{array}{rlrl}\sin x& =x-{\displaystyle \frac{x^3}{3!}}+{\displaystyle \frac{x^5}{5!}}-{\displaystyle \frac{x^7}{7!}}+\cdots & & =\sum _{n=0}^{\mathrm{\infty }}{\displaystyle \frac{(-1{)}^n{x}^{2n+1}}{(2n+1)!}}\\ \cos x& =1-{\displaystyle \frac{x^2}{2!}}+{\displaystyle \frac{x^4}{4!}}-{\displaystyle \frac{x^6}{6!}}+\cdots & & =\sum _{n=0}^{\mathrm{\infty }}{\displaystyle \frac{(-1{)}^n{x}^{2n}}{(2n)!}}\\ \tan x& =x+{\displaystyle \frac{x^3}{3}}+{\displaystyle \frac{2x^5}{15}}+{\displaystyle \frac{17x^7}{315}}+{\displaystyle \frac{62x^9}{2835}}\cdots \end{array}$

Complex Exponential Function

$e^{ix}=\cos x+i\sin x=\mathrm{cis}(\theta )$

$\begin{array}{rl}\cos x& ={\displaystyle \frac{e^{ix}+e^{-ix}}{2}}\\ \sin x& ={\displaystyle \frac{e^{ix}-e^{-ix}}{2i}}\end{array}$

Sources

• Wikipedia: Trigonometric Functions
• Wikipedia: List of Trigonometric Identities

Special Values

Common & Sub-Angle Values

Sources

• Wikipedia: Exact Trigonometric Values

Indefinite Integration

Basic Formulas

1. Power Rule

$(n\ne -1)$

${\displaystyle \int (f(x){)}^n{f}^{\prime }(x)dx={\displaystyle \frac{(f(x){)}^{n+1}}{n+1}}+C}$

${\displaystyle \int x^{n}dx={\displaystyle \frac{x^{n+1}}{n+1}}+C}$

${\displaystyle \int |x{|}^{n}dx={\displaystyle \frac{x|x{|}^n}{n+1}}+C}$

2. Logarithmic Integration

${\displaystyle \int {\displaystyle \frac{f^{\prime }(x)}{f(x)}}dx=\ln |f(x)|+C}$

${\displaystyle \int {\displaystyle \frac{1}{x}}dx=\ln |x|+C}$

3. Trigonometric Functions

$\begin{array}{rlrl}{\displaystyle \int \sin xdx}& & & =-\cos x+C\\ {\displaystyle \int \cos xdx}& & & =\sin x+C\\ {\displaystyle \int \tan xdx}& & & =-\ln |\cos x|+C& & =\ln |\sec x|+C\\ {\displaystyle \int \cot xdx}& & & =\ln |\sin x|+C& & =-\ln |\csc x|+C\\ {\displaystyle \int \sec xdx}& & & =\ln |\sec x+\tan x|+C& & =\ln |\tan (\frac{\pi }{4}+\frac{x}{2})|+C\\ {\displaystyle \int \csc xdx}& & & =\ln |\csc x-\cot x|+C& & =\ln |\tan \frac{x}{2}|+C\end{array}$

4. Exponential Function

${\displaystyle \int e^x(f(x)+f^{\prime }(x))dx=e^{x}f(x)+C}$

${\displaystyle \int a^{x}dx={\displaystyle \frac{a^x}{\ln a}}+C}$

${\displaystyle \int e^{x}dx=e^x+C}$

5. Special

$\begin{array}{rl}{\displaystyle \int {\displaystyle \frac{dx}{a^2+x^2}}}& ={\displaystyle \frac{1}{a}}{\tan }^{-1}{\displaystyle \frac{x}{a}}+C\\ {\displaystyle \int {\displaystyle \frac{dx}{a^2-x^2}}}& ={\displaystyle \frac{1}{2a}}\ln \left|{\displaystyle \frac{a+x}{a-x}}\right|+C\\ {\displaystyle \int {\displaystyle \frac{dx}{\sqrt{a^2-x^2}}}}& ={\sin }^{-1}{\displaystyle \frac{x}{a}}+C\\ {\displaystyle \int {\displaystyle \frac{dx}{\sqrt{x^2+a^2}}}}& =\ln |x+\sqrt{x^2+a^2}|+C\\ {\displaystyle \int {\displaystyle \frac{dx}{\sqrt{x^2-a^2}}}}& =\ln |x+\sqrt{x^2-a^2}|+C\\ {\displaystyle \int {\displaystyle \frac{dx}{x\sqrt{x^2-a^2}}}}& ={\displaystyle \frac{1}{a}}{\sec }^{-1}{\displaystyle \frac{x}{a}}+C\\ {\displaystyle \int \sqrt{a^2-x^2}dx}& ={\displaystyle \frac{x}{2}}\sqrt{a^2-x^2}+{\displaystyle \frac{a^2}{2}}{\sin }^{-1}{\displaystyle \frac{x}{a}}+C\\ {\displaystyle \int \sqrt{x^2+a^2}dx}& ={\displaystyle \frac{x}{2}}\sqrt{x^2+a^2}+{\displaystyle \frac{a^2}{2}}\ln |x+\sqrt{x^2+a^2}|+C\\ {\displaystyle \int \sqrt{x^2-a^2}dx}& ={\displaystyle \frac{x}{2}}\sqrt{x^2-a^2}-{\displaystyle \frac{a^2}{2}}\ln |x+\sqrt{x^2-a^2}|+C\end{array}$