Approximations

Irrational

Ratios and Angles

Basic Formulae

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

$\begin{aligned} \sin x & = \cos \left(90^{\circ }-x\right) & = {\dfrac {1}{\csc x}} \\ \cos x & = \sin \left(90^{\circ }-x\right) & = {\dfrac {1}{\sec x}} \\ \tan x & = \cot \left(90^{\circ }-x\right) & = {\dfrac {\sin x}{\cos x}} = {\dfrac {1}{\cot x}} \end{aligned}$

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

$\begin{aligned} \sin(A\pm B) & = \sin A\cos B\pm \cos A\sin B \\[3mu] \cos(A\pm B) & = \cos A\cos B\mp \sin A\sin B \end{aligned}$

$\begin{aligned} \tan\left(A\pm B\right) & = \dfrac{\tan A\pm\tan B}{1\mp\tan A\tan B} \\[10mu] \cot\left(A\pm B\right) & = \dfrac{\cot A\cot B\mp 1}{\cot B\pm \cot A} \end{aligned}$

$\begin{aligned} \sin\left(A+B\right)\sin\left(A-B\right) && = \sin^2A-\sin^2B && = \cos^2B-\cos^2A \\ \cos\left(A+B\right)\cos\left(A-B\right) && = \cos^2A-\sin^2B && = \cos^2B-\sin^2A \end{aligned}$

Three Angle

$\begin{aligned} \sin\left(A+B+C\right) & = \sin A\cos B\cos C+\cos A\sin B\cos C+\cos A\cos B\sin C-\sin A\sin B\sin C \\[3mu] & = \cos A\cos B\cos C\left(\tan A+\tan B+\tan C-\tan A\tan B\tan C\right) \end{aligned}$

$\begin{aligned} \cos\left(A+B+C\right) & = \cos A\cos B\cos C-\sin A\sin B\cos C-\sin A\cos B\sin C-\cos A\sin B\sin C \\[3mu] & = \cos A\cos B\cos C(1-\tan A\tan B-\tan B\tan C-\tan C\tan A) \end{aligned}$

$\begin{aligned} \tan\left(A+B+C\right) & = \dfrac{\tan A+\tan B+\tan C-\tan A\tan B\tan C}{1-\tan A\tan B-\tan B\tan C-\tan C\tan A} \\[12mu] \cot\left(A+B+C\right) & = \dfrac{\cot A+\cot B+\cot C-\cot A\cot B\cot C}{1-\cot A\cot B-\cot B\cot C-\cot C\cot A} \end{aligned}$

General

$\begin{aligned} S_{0} & = 1 \\ S_{1} & = \sum\limits_{i} x_{i} && = \sum\limits_{i} \tan \theta_{i} \\ S_{2} & = \sum\limits_{i < j} x_{i} x_{j} && = \sum\limits_{i < j} \tan \theta_{i} \tan \theta_{j} \\ S_{3} & = \sum\limits_{i < j < k} x_{i} x_{j} x_{k} && = \sum\limits_{i < j < k} \tan \theta_{i} \tan \theta_{j} \tan \theta_{k} \\ & \quad \vdots && \quad \vdots \end{aligned}$

Then,

$\begin{aligned} \tan\left(\theta_1+\theta_2+\cdots+\theta_n\right) & = \dfrac{S_1-S_3+S_5-S_7+\cdots}{1-S_2+S_4-S_6+\cdots} \\[16mu] \sec\left(\theta_1+\theta_2+\cdots+\theta_n\right) & = {\dfrac {\sec \theta_1 \sec \theta_2 \cdots \sec \theta_n}{S_{0}-S_{2}+S_{4}-\cdots }} \\[16mu] \csc\left(\theta_1+\theta_2+\cdots+\theta_n\right) & = {\dfrac {\sec \theta_1 \sec \theta_2 \cdots \sec \theta_n}{S_{1}-S_{3}+S_{5}-\cdots }} \end{aligned}$

$\cos2^r\theta\cdot\cos2^{r+1}\theta\cdot\cos2^{r+2}\theta\cdots\cos2^n\theta= \dfrac{\sin2^{n+1}\theta}{2^{n-r+1}\sin2^r\theta}$

$\begin{aligned} \sin(\alpha)+\sin(\alpha+\beta)+\sin(\alpha+2\beta)+\cdots+\sin\left(\alpha+n\beta\right) & = \dfrac{\sin \left((n+1)\frac{\beta}{2}\right)}{\sin\left(\frac{\beta}{2}\right)}\sin\left(\alpha+n\frac{\beta}{2}\right) \\[16mu] \cos(\alpha)+\cos(\alpha+\beta)+\cos(\alpha+2\beta)+\cdots+\cos\left(\alpha+n\beta\right) & = \dfrac{\sin\left((n+1)\frac{\beta}{2}\right)}{\sin\left(\frac{\beta}{2}\right)}\cos\left(\alpha+n\frac{\beta}{2}\right) \end{aligned}$

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

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

$\dfrac{\sin{\theta_1} \pm \sin{\theta_2} + \sin{\theta_3} + \cdots + \sin{\theta_n}}{\cos{\theta_1} \pm \cos{\theta_2} + \cos{\theta_3} + \cdots + \cos{\theta_n}} = \tan{\dfrac{\theta_1 \pm \theta_2 + \theta_3 + \cdots + \theta_n}{n} }$

Product to Sum

$\begin{aligned} 2\sin A\cos B & = \sin\left(A+B\right)+\sin\left(A-B\right) \\[6mu] 2\cos A\sin B & = \sin\left(A+B\right)-\sin\left(A-B\right) \\[6mu] 2\cos A\cos B & = \cos\left(A+B\right)+\cos\left(A-B\right) \\[6mu] 2\sin A\sin B & = \cos\left(A-B\right)-\cos\left(A+B\right) \end{aligned}$

Sum to Product

$\begin{aligned} \sin C+\sin D & = 2\sin\frac{C+D}2\cos\frac{C-D}2 \\[8mu] \sin C-\sin D & = 2\cos\frac{C+D}2\sin\frac{C-D}2 \\[8mu] \cos C+\cos D & = 2\cos\frac{C+D}2\mathrm{cos}\frac{C-D}2 \\[8mu] \cos C-\cos D & = -2\sin\frac{C+D}2\sin\frac{C-D}2 \end{aligned}$

Multiple Angle

Two/Three Angle

$\begin{aligned} \sin2\theta& = 2\sin \theta\cos \theta \\[3mu] & = \dfrac{2\tan \theta}{1+\tan^2\theta} \end{aligned}$

$\begin{aligned} \cos2\theta& = \cos^2\theta-\sin^2\theta \\[3mu] & = 2\cos^2\theta-1 \\[3mu] & = 1-2\sin^2\theta \\[3mu] & = \dfrac{1-\tan^2\theta}{1+\tan^2\theta} \end{aligned}$

$\tan2\theta= \dfrac{2\tan\theta}{1-\tan^2\theta}$

$\begin{aligned} \sin3\theta & = 3\sin \theta-4\sin^3\theta \\[3mu] & = 4\sin\left(60^\circ-\theta\right)\sin (\theta)\sin\left(60^\circ+\theta\right) \end{aligned}$

$\begin{aligned} \cos3\theta & = 4\cos^3\theta-3\cos \theta \\[3mu] & = 4\cos\left(60^\circ-\theta\right)\cos (\theta)\cos\left(60^\circ+\theta\right) \end{aligned}$

$\begin{aligned} \tan3\theta & = \dfrac{3\tan \theta-\tan^3\theta}{1-3\tan^2\theta} \\[3mu] & = \tan\left(60^{\circ}-\theta\right)\tan (\theta)\tan\left(60^{\circ}+\theta\right) \end{aligned}$

Multi(>3) Angle

$\begin{aligned} \sin4\theta & = 4\sin \theta\cos^3 \theta - 4\cos \theta\sin^3 \theta \\ \cos4\theta & = 8\cos^4\theta-8\cos^2\theta+1 \\ \tan4\theta & = \dfrac{4\tan\theta-4\tan^3\theta}{1-6\tan^2\theta+\tan^4\theta} \\ \sin5\theta & = 16\sin^5\theta-20\sin^3\theta+5\sin \theta \\ \cos5\theta & = 16\cos^5\theta-20\cos^3\theta+5\cos \theta \end{aligned}$

Half Angle

$\begin{aligned} \sin {\dfrac {\theta }{2}} & = \operatorname {sgn} \left(\sin {\dfrac {\theta }{2}}\right){\sqrt {\dfrac {1-\cos \theta }{2}}} \\ \cos {\dfrac {\theta }{2}} & = \operatorname {sgn} \left(\cos {\dfrac {\theta }{2}}\right){\sqrt {\dfrac {1+\cos \theta }{2}}} \end{aligned}$

$\begin{aligned} \tan {\dfrac {\theta }{2}} & ={\frac {1-\cos \theta }{\sin \theta }}\\[8mu] & ={\frac {\sin \theta }{1+\cos \theta }}\\[8mu] & =\csc \theta -\cot \theta \\[1mu] & ={\frac {\tan \theta }{1+\sec {\theta }}}\\[1mu] & =\operatorname {sgn}(\sin \theta ){\sqrt {\frac {1-\cos \theta }{1+\cos \theta }}} \end{aligned}$

Special Cases

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

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

Power Reduction

Special Cases

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

$\begin{aligned} \sin2A+\sin2B+\sin2C & = 4\sin A\sin B\sin C \\ \sin2A+\sin2B-\sin2C & = 4\cos A\cos B\sin C \\ \cos2A+\cos2B+\cos2C & = -1-4\cos A\cos B\cos C \\ \cos2A+\cos2B-\cos2C & = 1-4\sin A\sin B\cos C \\[20mu] \sin A+\sin B+\sin C & =4\cos\frac A2\cos\frac B2\cos\frac C2 \\ \sin A+\sin B-\sin C & =4\sin\frac A2\sin\frac B2\cos\frac C2 \\ \cos A+\cos B+\cos C & =1+4\sin\frac A2\sin\frac B2\sin\frac C2 \\ \cos A+\cos B-\cos C & =-1+4\cos\frac A2\cos\frac B2\sin\frac C2 \\[20mu] \sin^2A+\sin^2B-\sin^2C & = 2\sin A\sin B\cos C \\ \cos^2A+\cos^2B-\cos^2C & = 1-2\sin A\sin B\cos C \\ \cos^2A+\cos^2B+\cos^2C & = 1-2\cos A\cos B\cos C \\ \sin^2A+\sin^2B+\sin^2C & = 2+2\cos A\cos B\cos C \\[20mu] \tan A+\tan B+\tan C & = \tan A\tan B\tan C \\ \cot B\cot C+\cot C\cot A+\cot A\cos B & = 1 \\ \tan\frac B2\tan\frac C2+\tan\frac C2\tan\frac A2+\tan\frac A2.\tan\frac B2 & = 1 \\ \cot\frac A2+\cot\frac B2+\cot\frac C2 & = \cot\frac A2\cot\frac B2\cot\frac C2 \end{aligned}$

Series Expansion

$\begin{aligned} \sin x & = x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dfrac{x^7}{7!}+\cdots && = \sum\limits_{n=0}^\infty\dfrac{(-1)^nx^{2n+1}}{(2n+1)!} \\ \cos x & = 1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\dfrac{x^6}{6!}+\cdots && = \sum\limits_{n=0}^\infty\dfrac{(-1)^nx^{2n}}{(2n)!} \\ \tan x & = x+\dfrac{x^{3}}{3}+\dfrac{2x^{5}}{15}+\dfrac{17x^{7}}{315}+\dfrac{62x^{9}}{2835}\cdots \end{aligned}$

Complex Exponential Function

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

$\begin{aligned} \cos x & = {\dfrac {e^{ix}+e^{-ix}}{2}} \\ \sin x & = {\dfrac {e^{ix}-e^{-ix}}{2i}} \end{aligned}$

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

$\quad (n\not ={-1})$

$\displaystyle\int (f(x))^n f'(x) \, dx = \dfrac{(f(x))^{n+1}}{n+1} + C$

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

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

2. Logarithmic Integration

$\displaystyle\int \dfrac{f'(x)}{f(x)} \, dx = \ln |f(x)| + C$

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

3. Trigonometric Functions

$\begin{aligned} \displaystyle\int \sin x \, dx &&& = -\cos x + C \\ \displaystyle\int \cos x \, dx &&& = \sin x + C \\ \displaystyle\int \tan x \, dx &&& = -\ln |\cos x| + C && = \ln |\sec x| + C \\ \displaystyle\int \cot x \, dx &&& = \ln |\sin x| + C && = -\ln |\csc x| + C \\ \displaystyle\int \sec x \, dx &&& = \ln |\sec x + \tan x| + C && = \ln \left| \tan \left(\frac{\pi}{4} + \frac{x}{2}\right) \right| + C \\ \displaystyle\int \csc x \, dx &&& = \ln |\csc x - \cot x| + C && = \ln \left| \tan \frac{x}{2} \right| + C \end{aligned}$

4. Exponential Function

$\displaystyle\int e^x(f(x)+f'(x)) \, dx = e^x f(x) + C$

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

$\displaystyle\int e^x \, dx = e^x + C$

5. Special

$\begin{aligned} \displaystyle\int \dfrac{dx}{a^2 + x^2} & = \dfrac{1}{a} \tan^{-1} \dfrac{x}{a} + C \\[8mu] \displaystyle\int \dfrac{dx}{a^2 - x^2} & = \dfrac{1}{2a} \ln \left| \dfrac{a+x}{a-x} \right| + C \\[25mu] \displaystyle\int \dfrac{dx}{\sqrt{a^2 - x^2}} & = \sin^{-1} \dfrac{x}{a} + C \\[25mu] \displaystyle\int \dfrac{dx}{\sqrt{x^2 + a^2}} & = \ln \left|x + \sqrt{x^2 + a^2}\right| + C \\[8mu] \displaystyle\int \dfrac{dx}{\sqrt{x^2 - a^2}} & = \ln \left| x + \sqrt{x^2 - a^2} \right| + C \\[25mu] \displaystyle\int \dfrac{dx}{x\sqrt{x^2 - a^2}} & = \dfrac{1}{a} \sec^{-1} \dfrac{x}{a} + C \\[25mu] \displaystyle\int \sqrt{a^2 - x^2} \, dx & = \dfrac{x}{2} \sqrt{a^2 - x^2} + \dfrac{a^2}{2} \sin^{-1} \dfrac{x}{a} + C \\[8mu] \displaystyle\int \sqrt{x^2 + a^2} \, dx & = \dfrac{x}{2} \sqrt{x^2 + a^2} + \dfrac{a^2}{2} \ln \left|x + \sqrt{x^2 + a^2}\right| + C \\[8mu] \displaystyle\int \sqrt{x^2 - a^2} \, dx & = \dfrac{x}{2} \sqrt{x^2 - a^2} - \dfrac{a^2}{2} \ln \left|x + \sqrt{x^2 - a^2}\right| + C \\[8mu] \end{aligned}$