Inverse Trigonometric Functions

• The inverse trigonometric functions are:

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

• The derivatives of the inverse trigonometric functions are:

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

• The integrals of the inverse trigonometric functions are:

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

• 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 $$\sin^{-1}\frac{1}{2}=\frac{\pi}{6}$$

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

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

• Find the value of $\tan^{-1}1$.

Solution $$\tan^{-1}1=\frac{\pi}{4}$$

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

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

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

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

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

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

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

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

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

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

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

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

• Find the integral of $\sin^{-1}x$.

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

• Find the integral of $\cos^{-1}x$.

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

• Find the integral of $\tan^{-1}x$.

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