$$\color{Blue}{\textbf{Calculus}}$$
$$\color{Magenta}{\textbf{1.Trigonometry}}$$
$\textbf{Pythagorean Identities:}$
- $\sin^{2}\theta + \cos^{2}\theta = 1$
- $1+\tan^{2}\theta = \sec^{2}\theta$
- $1+\cot^{2}\theta = \csc^{2}\theta$
$\textbf{Reciprocal Identities:}$
- $\sin\theta = \dfrac{1}{\csc\theta}$
- $\cos\theta = \dfrac{1}{\sec\theta}$
- $\tan\theta = \dfrac{1}{\cot\theta}$
- $\csc\theta = \dfrac{1}{\sin\theta}$
- $\sec\theta = \dfrac{1}{\cos\theta}$
- $\cot\theta = \dfrac{1}{\tan\theta}$
$\textbf{Tangent and Cotangent Identities:}$
- $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$
- $\cot\theta = \dfrac{\cos\theta}{\sin\theta}$
$\textbf{Formulas for twice of angle:}$
- $\sin2\theta = 2 \sin\theta \cos\theta$
- $\cos2\theta = 2\cos^{2}\theta\: – 1 = 1 – 2\sin^{2}\theta $
- $\tan2\theta = \dfrac{2\tan\theta} {1− \tan^{2}\theta} = \dfrac{\sin 2\theta} {1− 2\sin^{2}\theta}$
-------------------------------------------------------------------------------------------------------------------
$\color{\Purple}{\textbf{Some important things, we should know}}$
- ${\color{Red} {\cos 0 = 1,\cos \pi = -1,\cos 2\pi = 1,\dots}}$
- ${\color{Blue}{ \text{In general}\: \cos n\pi = (-1)^{n}\: \text{where}\: n=0,1,2,\dots}}$
- ${\color{Magenta} {\sin 0 = 0,\sin \pi = 0,\sin 2\pi = 0,\dots}}$
- ${\color{Green} {\text{In general}\: \sin n\pi = 0\: \text{where}\: n=0,1,2,\dots}}$
- ${\color{Orange} {\sin(\pi-x)=\sin x,\sin (2\pi-x)=-\sin x,\sin(3\pi-x)=\sin x}}$
- ${\color{Teal} {\text{In general}\: \sin (n\pi-x)=(-1)^{n+1}\sin x\: \text{ where} \: n=0,1,2,\dots}}$
- ${\color{Orchid} {\cos(\pi-x)=-\cos x,\cos (2\pi-x)=\cos x,\cos (3\pi-x)=-\cos x}}$
- ${\color{purple} {\text{ In general}\: \cos (n\pi-x)=(-1)^{n}\cos x \: \text{where}\: n=0,1,2,\dots}}$
__________________________________________________________
$f(x) = \sin x$
$f(x) = \cos x$
Visualization:
$$\color{Red}{\textbf{2.Integral & Differential Calculus}}$$
$${\color{Teal}{2.1.\textbf{Some Useful Formulae for Differentiation}}}$$
- $\dfrac{\mathrm{d} }{\mathrm{d} x}(c) = 0;c = \text{Constant}$
- $\dfrac{\mathrm{d} }{\mathrm{d} x}(x) = 1$
- $\dfrac{\mathrm{d} }{\mathrm{d} x}(x^{n}) = nx^{n-1}$
- $\dfrac{\mathrm{d} }{\mathrm{d} x}(e^{x}) = e^{x}$
- $\dfrac{\mathrm{d} }{\mathrm{d} x}(a^{x}) = a^{x}\log_{e}a$
- $\dfrac{\mathrm{d} }{\mathrm{d} x}(\log x) = \dfrac{1}{x}$
- $\dfrac{\mathrm{d} }{\mathrm{d} x}(\log_{a} x) = \dfrac{1}{x}\log_{a}e$
- $\dfrac{\mathrm{d} }{\mathrm{d} x}(\sin x) = \cos x$
- $\dfrac{\mathrm{d} }{\mathrm{d} x}(\cos x) = -\sin x$
- $\dfrac{\mathrm{d} }{\mathrm{d} x}(\tan x) = \sec^{2} x$
- $\dfrac{\mathrm{d} }{\mathrm{d} x}(\cot x) = -\csc^{2} x$
- $\dfrac{\mathrm{d} }{\mathrm{d} x}(\csc x) = \csc x \cot x$
- $\dfrac{\mathrm{d} }{\mathrm{d} x}(\sec x) = \sec x\tan x$
- $\dfrac{\mathrm{d} }{\mathrm{d} x}(\sin^{-1} x) = \dfrac{1}{\sqrt{1-x^{2}}}$
- $\dfrac{\mathrm{d} }{\mathrm{d} x}(\cos^{-1} x) = \dfrac{-1}{\sqrt{1-x^{2}}}$
- $\dfrac{\mathrm{d} }{\mathrm{d} x}(\tan^{-1} x) = \dfrac{1}{1 + x^{2}}$
- $\dfrac{\mathrm{d} }{\mathrm{d} x}(\cot^{-1} x) = \dfrac{-1}{1 + x^{2}}$
- $\dfrac{\mathrm{d} }{\mathrm{d} x}(\sec^{-1} x) = \dfrac{1}{x \sqrt{x^{2}-1 }}$
- $\dfrac{\mathrm{d} }{\mathrm{d} x}(\csc^{-1} x) = \dfrac{-1}{x \sqrt{x^{2}-1}}$
- $\dfrac{\mathrm{d} }{\mathrm{d} x}(\sqrt{x}) = \dfrac{1}{ 2 \sqrt{x}}$
$${\color{Purple}{2.2.\textbf{Some Useful Formulae for Integration}}}$$
$\textbf{Algebraic Functions}$
- $\displaystyle{}\int k\: dx = kx + C$
- $\displaystyle{}\int x^{n}\: dx = \dfrac{x^{n+1}}{n+1} + C$
- $\displaystyle{}\int (ax + b)^{n}\: dx = \dfrac{(ax + b)^{n+1}}{a(n+1)} + C\:;(\text{for}\: n \neq -1)$
- $\displaystyle{}\int \dfrac{1}{x}\: dx = \ln x + C\:;\text{(for positive values of $x$ only)}$
- $\displaystyle{}\int \dfrac{c}{ax + b}\: dx = \dfrac{c}{a}\ln(ax +b) + C$
$\textbf{Exponential Functions}$
- $\displaystyle{}\int e^{x} dx = e^{x}+ C$
- $\displaystyle{}\int a^{x} dx = \dfrac{a^{x}}{\ln a}+ C$
$\textbf{Logarithm Functions}$
- $\displaystyle{}\int \ln x\: dx = x\ln x\: – x+ C$
- $\displaystyle{}\int \log_{a} x\: dx = x\log_{a} x\: – \dfrac{x}{\log a} + C$
$\textbf{Trigonometric Functions}$
$\displaystyle{}\int \sin x\: dx = -\cos x + C$