Inverse Trigonometric Functions Practice Problems

A collection of practice flashcards covering inverse trigonometric identities, principal values, and simplification problems from the lecture notes.

What is the constant value of the expression $tan^{-1} x + cot^{-1} x$?

$$\frac{\pi}{2}$$

Find the value of $sin^{-1}(sin \frac{3\pi}{5})$.

$$\frac{2\pi}{5}$$

According to question 20, what is the value of $cot(tan^{-1} x + cot^{-1} x)$?

$$0$$

Write the values of $x$ for which the equation $2 tan^{-1} x = cos^{-1} \frac{1-x^2}{1+x^2}$ holds.

$$x \ge 0$$

Find the value of $sin^{-1}(sin \frac{2\pi}{3})$.

$$\frac{\pi}{3}$$

What is the value of $cos(sec^{-1} x + cosec^{-1} x)$ for $|x| \ge 1$?

$$0$$

What is the domain of the function $f(x) = sin^{-1} x$?

$$[-1,1]$$

Find the principal value of $cot^{-1}(-\frac{1}{\sqrt{3}})$.

$$\frac{2\pi}{3}$$

What is the principal value of $cos^{-1}(-\frac{1}{2})$?

$$\frac{2\pi}{3}$$

Find the principal value of $cosec^{-1}(-\sqrt{2})$.

$$-\frac{\pi}{4}$$

What is the value of $cos(tan^{-1} \frac{3}{4})$?

$$4/5$$

Write the function $tan^{-1}\left(\frac{\cos x - \sin x}{\cos x + \sin x}\right)$ in simplest form where $0 < x < \pi$.

$$\frac{\pi}{4} - x$$

Show that $2 sin^{-1} \frac{3}{5}$ is equal to what value in terms of $tan^{-1}$?

$$tan^{-1} \frac{24}{7}$$

Write the function $cot^{-1}\left(\frac{1}{\sqrt{x^2 - 1}}\right)$ where $x > 1$ in the simplest form.

$$sec^{-1} x$$

Show that $sin^{-1}(2x\sqrt{1-x^2}) = 2sin^{-1} x$ using which substitution for $x$?

$$x = sin \theta$$

If $sin(sin^{-1} \frac{1}{5} + cos^{-1} x) = 1$, solve for $x$.

$$1/5$$

Write the simplest form of the function $tan^{-1}\left(\frac{\sqrt{1+x^2} - 1}{x}\right)$ where $x \neq 0$.

$$\frac{1}{2} tan^{-1} x$$

Find the value of $tan^{-1}(\sqrt{3}) - sec^{-1}(-2)$.

$$-\frac{\pi}{3}$$