Mathematics - All Important Notes, Formula and Concept

Inverse Trigonometric function - Notes, Concept and All Important Formula

INVERSE TRIGONOMETRIC FUNCTION 1. DOMAIN, RANGE & GRAPH OFINVERSE TRIGONOMETRIC FUNCTIONS : (a) \(\mathrm{f}^{-1}:[-1,1] \rightarrow[-\pi / 2, \pi / 2]\) , \(\mathrm{f}^{-1}(\mathrm{x})=\sin ^{-1}(\mathrm{x})\) (b) \(\mathrm{f}^{-1}:[-1,1] \rightarrow[0, \pi]\) , \(\mathrm{f}^{-1}(\mathrm{x})=\cos ^{-1} \mathrm{x}\) (c) \(\mathrm{f}^{-1}: \mathrm{R} \rightarrow(-\pi / 2, \pi / 2)\) , \(\mathrm{f}^{-1}(\mathrm{x})=\tan ^{-1} \mathrm{x}\) (d) \(\mathrm{f}^{-1}: \mathrm{R} \rightarrow(0, \pi)\) \(\mathrm{f}^{-1}(\mathrm{x})=\cot ^{-1} \mathrm{x}\) (e) \(\mathrm{f}^{-1}:(-\infty,-1] \cup[1, \infty)\) \(\rightarrow[0, \pi / 2) \cup(\pi / 2, \pi]\) , \(\mathrm{f}^{-1}(\mathrm{x})=\sec ^{-1} \mathrm{x}\) (f) \(\mathrm{f}^{-1}:(-\infty,-1] \cup[1, \infty)\) \(\rightarrow[-\pi / 2,0) \cup(0, \pi / 2]\) \(\mathrm{f}^{-1}(\mathrm{x})=\operatorname{cosec}^{-1} \mathrm{x}\) All Chapter Notes, Concept and Important Formula 2. PROPERTIES OF INVERSE CIRCULAR FUNCTIONS: Property-1 : (i) \(y=\sin ...

Trigonometry Ratios and Identities - Notes, Concept and All Important Formula

TRIGONOMETRIC RATIOS & IDENTITIES Table Of Contents 1. RELATION BETWEEN SYSTEM OF MEASUREMENT OF ANGLES : \(\dfrac{D}{90}=\dfrac{G}{100}=\dfrac{2 C}{\pi}\) 1 Radian \(=\dfrac{180}{\pi}\) degree \(\approx 57^{\circ} 17^{\prime} 15^{\prime \prime}\) (approximately) 1 degree \(=\dfrac{\pi}{180}\) radian \(\approx 0.0175\) radian All Chapter Notes, Concept and Important Formula 2. BASIC TRIGONOMETRIC IDENTITIES : (a) \(\sin ^{2} \theta+\cos ^{2} \theta=1\) or \(\sin ^{2} \theta=1-\cos ^{2} \theta\) or \(\cos ^{2} \theta=1-\sin ^{2} \theta\) (b) \(\sec ^{2} \theta-\tan ^{2} \theta=1\) or \(\sec ^{2} \theta=1+\tan ^{2} \theta\) or \(\tan ^{2} \theta=\sec ^{2} \theta-1\) (c) If \(\sec \theta+\tan \theta\) \(=\mathrm{k} \Rightarrow \sec \theta-\tan \theta\) \(=\dfrac{1}{\mathrm{k}} \Rightarrow 2 \sec \theta\) \(=\mathrm{k}+\dfrac{1}{\mathrm{k}}\) (d) \(\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1\) or \(\operatorname{cosec}^{2} \theta=1+\cot ^{2} \th...

Difference and relation between Differentiation and Integration

Relation between Differentiation and Integration Table Of Contents Look at the information given below. \[\mathbf{ y=f(x)}\] \[ \mathbf{ f'(x)\rightarrow \text{Derivatives of f(x)}}\] \[ \mathbf{\displaystyle \int_a^b f’(x) = ?}\] Can you tell me the value of above integral? Yes, it will be equal to f(b)-f(a) . We have already known this result. It tells us that integration is just the reverse of differentiation, integral of the derivative of the function f(x) is just equal to the difference in the function f(x) evaluated at the limits of integration. Indefinite integration- Notes and Formula Part 1 Now with this topic, we will understand how to apply this result to find the integral of a function. Consider this function \(\mathbf{g(x)=x^2}\) Let's find the integral of this function from a to b i.e \(\mathbf{\displaystyle \int_a^b g(x) \, dx}\) . Can you think how we can apply this result to find ...

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