SATHEE: Shortcut Methods

Shortcut Methods

Inverse Trigonometric Functions

The inverse trigonometric functions are:

$\sin^{- 1} x : [- 1, 1] \to [- \frac{π}{2}, \frac{π}{2}]$ $\cos^{- 1} x : [- 1, 1] \to [0, π]$ $\tan^{- 1} x : All real numbers \to (- \frac{π}{2}, \frac{π}{2})$

The derivatives of the inverse trigonometric functions are:

$\frac{d}{d x} \sin^{- 1} x = \frac{1}{\sqrt{1 - x^{2}}}$ $\frac{d}{d x} \cos^{- 1} x = - \frac{1}{\sqrt{1 - x^{2}}}$ $\frac{d}{d x} \tan^{- 1} x = \frac{1}{1 + x^{2}}$

The integrals of the inverse trigonometric functions are:

$\int \sin^{- 1} x d x = x \sin^{- 1} x - \sqrt{1 - x^{2}}$ $\int \cos^{- 1} x d x = x \cos^{- 1} x - \sqrt{1 - x^{2}}$ $\int \tan^{- 1} x d x = x \tan^{- 1} x - \frac{1}{2} \ln (1 + x^{2})$

The inverse trigonometric functions can be used to solve a variety of equations, including:

$\sin^{- 1} x = y$ $\cos^{- 1} x = y$ $\tan^{- 1} x = y$

Typical Numericals

Find the value of $\sin^{- 1} \frac{1}{2}$ .

Solution $\sin^{- 1} \frac{1}{2} = \frac{π}{6}$

Find the value of $\cos^{- 1} \frac{\sqrt{3}}{2}$ .

Solution $\cos^{- 1} \frac{\sqrt{3}}{2} = \frac{π}{3}$

Find the value of $\tan^{- 1} 1$ .

Solution $\tan^{- 1} 1 = \frac{π}{4}$

Solve the equation $\sin^{- 1} x = \frac{π}{3}$ .

Solution $x = \frac{\sqrt{3}}{2}$

Solve the equation $\cos^{- 1} x = \frac{π}{4}$ .

Solution $x = \frac{1}{\sqrt{2}}$

Solve the equation $\tan^{- 1} x = \frac{π}{6}$ .

Solution $x = \frac{1}{\sqrt{3}}$

Find the derivative of $\sin^{- 1} x$ .

Solution $\frac{d}{d x} \sin^{- 1} x = \frac{1}{\sqrt{1 - x^{2}}}$

Find the derivative of $\cos^{- 1} x$ .

Solution $\frac{d}{d x} \cos^{- 1} x = - \frac{1}{\sqrt{1 - x^{2}}}$

Find the derivative of $\tan^{- 1} x$ .

Solution $\frac{d}{d x} \tan^{- 1} x = \frac{1}{1 + x^{2}}$

Find the integral of $\sin^{- 1} x$ .

Solution $\int \sin^{- 1} x d x = x \sin^{- 1} x - \sqrt{1 - x^{2}}$

Find the integral of $\cos^{- 1} x$ .

Solution $\int \cos^{- 1} x d x = x \cos^{- 1} x - \sqrt{1 - x^{2}}$

Find the integral of $\tan^{- 1} x$ .

Solution $\int \tan^{- 1} x d x = x \tan^{- 1} x - \frac{1}{2} \ln (1 + x^{2})$

Shortcut Methods

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Shortcut Methods

Table of Contents

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Mentorship

NCERT Exercise Video Solution

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