1.
Write the domain and principal value branch (range) of $y = \sin^{-1}x$.
2.
Write the domain and principal value branch (range) of $y = \cos^{-1}x$.
3.
Write the principal value branch (range) of $y = \tan^{-1}x$. How is it different from $\sin^{-1}x$?
4.
Write the principal value branch (range) of $y = \sec^{-1}x$. Which angle is excluded from this range?
5.
True or False: The inverse of a trigonometric function exists only if we restrict its domain to make it bijective (one-one and onto).
6.
True or False: $\sin^{-1}x$ is exactly the same as $(\sin x)^{-1}$.
7.
Find the principal value of $\sin^{-1}\left(\frac{1}{2}\right)$.
8.
Find the principal value of $\cos^{-1}\left(\frac{\sqrt{3}}{2}\right)$.
9.
Find the principal value of $\tan^{-1}(1)$.
10.
Find the principal value of $\sin^{-1}\left(-\frac{1}{2}\right)$.
11.
Find the principal value of $\cos^{-1}\left(-\frac{1}{2}\right)$.
12.
Find the principal value of $\tan^{-1}(-\sqrt{3})$.
13.
Evaluate: $\sin\left(\sin^{-1}\frac{1}{4}\right)$.
14.
Evaluate: $\cos^{-1}\left(\cos\frac{\pi}{6}\right)$.
15.
Express $\text{cosec}^{-1}(x)$ in terms of $\sin^{-1}$.
16.
Evaluate using reciprocal property: $\sec^{-1}(2)$. (Hint: relate it to $\cos^{-1}$).
17.
Complete the formula: $\sin^{-1}(-x) = $ _______________.
18.
Complete the formula: $\cos^{-1}(-x) = $ _______________.
19.
Complete the formula: $\tan^{-1}(-x) = $ _______________.
20.
What is the value of $\sin^{-1}(x) + \cos^{-1}(x)$? (Assume $x \in [-1, 1]$)
21.
What is the value of $\tan^{-1}(x) + \cot^{-1}(x)$?
22.
What is the value of $\text{cosec}^{-1}(x) + \sec^{-1}(x)$?
23.
Write the formula for $\tan^{-1}x + \tan^{-1}y$ when $xy < 1$.
24.
Write the formula for $\tan^{-1}x - \tan^{-1}y$ when $xy > -1$.
25.
Express $2\tan^{-1}x$ in terms of $\tan^{-1}$.
26.
Express $2\tan^{-1}x$ in terms of $\sin^{-1}$.
27.
Express $2\tan^{-1}x$ in terms of $\cos^{-1}$.
28.
To simplify an expression containing $\sqrt{a^2 - x^2}$, what is the most suitable substitution for $x$?
29.
To simplify an expression containing $\sqrt{a^2 + x^2}$, what is the most suitable substitution for $x$?
30.
To simplify an expression containing $\sqrt{x^2 - a^2}$, what is the most suitable substitution for $x$?
