Practice Questions

Inverse Trigonometric Functions

1
easySubjective

Justify whether the identity $\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}$ holds for $x=1.5$.

2
easySubjective

Calculate the principal value of cos⁡−1(−32)\cos^{-1}\left(-\frac{\sqrt{3}}{2}\right).

3
easySubjective

Determine the domain of the function f(x)=sin⁡−1(3x−1)f(x) = \sin^{-1}(3x-1).

4
easySubjective

Formulate an expression for $\csc^{-1}(x)$ in terms of $\sin^{-1}$ for $|x| \ge 1$.

5
mediumSubjective

Calculate the value of cos⁡−1(cos⁡(7π6))\cos^{-1}\left(\cos\left(\frac{7\pi}{6}\right)\right).

6
mediumSubjective

Simplify the expression tan⁡−1(1−cos⁡x1+cos⁡x)\tan^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right) for 0<x<π0 < x < \pi.

7
mediumSubjective

Solve the equation: tan⁡−1(2x)+tan⁡−1(3x)=π4\tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4}.

8
mediumSubjective

Analyze and simplify tan⁡−1(xa2−x2)\tan^{-1}\left(\frac{x}{\sqrt{a^2-x^2}}\right) for ∣x∣<a|x|<a.

9
mediumSubjective

Solve for xx: cos⁡(tan⁡−1x)=sin⁡(cot⁡−134)\cos(\tan^{-1} x) = \sin(\cot^{-1}\frac{3}{4}).

10
mediumSubjective

Demonstrate that cos⁡−145+tan⁡−135=tan⁡−12711\cos^{-1}\frac{4}{5} + \tan^{-1}\frac{3}{5} = \tan^{-1}\frac{27}{11}.

11
mediumSubjective

Prove that $\tan^{-1}\left(\frac{1}{5}\right) + \tan^{-1}\left(\frac{1}{7}\right) + \tan^{-1}\left(\frac{1}{3}\right) + \tan^{-1}\left(\frac{1}{8}\right) = \frac{\pi}{4}$.

12
mediumSubjective

Calculate the value of tan⁡−1(tan⁡7π6)\tan^{-1}(\tan \frac{7\pi}{6}).

13
mediumSubjective

Find the value of the expression sin⁡(π2−sin⁡−1(−12))\sin\left(\frac{\pi}{2} - \sin^{-1}\left(-\frac{1}{2}\right)\right).

14
mediumSubjective

Solve the equation sin⁡−15x+sin⁡−112x=π2\sin^{-1}\frac{5}{x} + \sin^{-1}\frac{12}{x} = \frac{\pi}{2}.

15
mediumSubjective

Critique the following step in a calculation: $\sin^{-1}(\sin(2\pi/3)) = 2\pi/3$.

16
mediumSubjective

Propose a trigonometric substitution for x to simplify the expression $\tan^{-1}\left(\frac{x}{\sqrt{a^2-x^2}}\right)$.

17
hardSubjective

Solve for xx: sin⁡−1x+sin⁡−1(2x)=π3\sin^{-1}x + \sin^{-1}(2x) = \frac{\pi}{3}.

18
hardSubjective

Evaluate the validity of the statement: The domain of $\cos^{-1}(3x-4)$ is $[1, 5/3]$.

19
hardSubjective

If cos⁡−1(xa)+cos⁡−1(yb)=α\cos^{-1}\left(\frac{x}{a}\right) + \cos^{-1}\left(\frac{y}{b}\right) = \alpha, demonstrate that x2a2−2xyabcos⁡α+y2b2=sin⁡2α\frac{x^2}{a^2} - \frac{2xy}{ab}\cos\alpha + \frac{y^2}{b^2} = \sin^2\alpha.

20
hardSubjective

Analyze and simplify the expression cot⁡−1(1+sin⁡x+1−sin⁡x1+sin⁡x−1−sin⁡x)\cot^{-1}\left(\frac{\sqrt{1+\sin x} + \sqrt{1-\sin x}}{\sqrt{1+\sin x} - \sqrt{1-\sin x}}\right) for x∈(0,π2)x \in \left(0, \frac{\pi}{2}\right).

21
hardSubjective

Calculate the value of sin⁡(2tan⁡−113)+cos⁡(tan⁡−1(22))\sin\left(2\tan^{-1}\frac{1}{3}\right) + \cos\left(\tan^{-1}(2\sqrt{2})\right).