Chapter 2

Inverse Trigonometric Functions

Every example and exercise of the chapter, worked in full. Each page gives the question, the complete solution, the reason behind every step, and the basics it rests on.

Interactive lesson All chapters Past paper questions (18)

Examples

Example 1Principal value of $\sin^{-1}\left(\frac{1}{\sqrt2}\right)$ worked
Example 2Principal value of $\cot^{-1}\left(\frac{-1}{\sqrt3}\right)$ worked
Example 3 $\sin^{-1}\left(2x\sqrt{1-x^2}\right)$ in terms of $\sin^{-1}x$ and $\cos^{-1}x$ worked
Example 4Simplest form of $\tan^{-1}\frac{\cos x}{1-\sin x}$ worked
Example 5Simplest form of $\cot^{-1}\frac{1}{\sqrt{x^2-1}}$ worked
Example 6The value of $\sin^{-1}\left(\sin\frac{3\pi}{5}\right)$ worked

Exercise 2.1

Exercise 2.1 · Q1 $\sin^{-1}\left(-\frac12\right)$ worked
Exercise 2.1 · Q2 $\cos^{-1}\left(\frac{\sqrt3}{2}\right)$ worked
Exercise 2.1 · Q3 $\operatorname{cosec}^{-1}(2)$ worked
Exercise 2.1 · Q4 $\tan^{-1}\left(-\sqrt3\right)$ worked
Exercise 2.1 · Q5 $\cos^{-1}\left(-\frac12\right)$ worked
Exercise 2.1 · Q6 $\tan^{-1}(-1)$ worked
Exercise 2.1 · Q7 $\sec^{-1}\left(\frac{2}{\sqrt3}\right)$ worked
Exercise 2.1 · Q8 $\cot^{-1}\left(\sqrt3\right)$ worked
Exercise 2.1 · Q9 $\cos^{-1}\left(-\frac{1}{\sqrt2}\right)$ worked
Exercise 2.1 · Q10 $\operatorname{cosec}^{-1}\left(-\sqrt2\right)$ worked
Exercise 2.1 · Q11 $\tan^{-1}1+\cos^{-1}\left(-\frac12\right)+\sin^{-1}\left(-\frac12\right)$ worked
Exercise 2.1 · Q12 $\cos^{-1}\left(\frac12\right)+2\sin^{-1}\left(\frac12\right)$ worked
Exercise 2.1 · Q13 (MCQ)If $\sin^{-1}x=y$ ⁠, where can $y$ lie?worked
Exercise 2.1 · Q14 (MCQ) $\tan^{-1}\sqrt3-\sec^{-1}(-2)$ worked

Exercise 2.2

Exercise 2.2 · Q1 $3\sin^{-1}x=\sin^{-1}(3x-4x^3)$ worked
Exercise 2.2 · Q2 $3\cos^{-1}x=\cos^{-1}(4x^3-3x)$ worked
Exercise 2.2 · Q3Simplest form of $\tan^{-1}\frac{\sqrt{1+x^2}-1}{x}$ worked
Exercise 2.2 · Q4Simplest form of $\tan^{-1}\sqrt{\frac{1-\cos x}{1+\cos x}}$ worked
Exercise 2.2 · Q5Simplest form of $\tan^{-1}\frac{\cos x-\sin x}{\cos x+\sin x}$ worked
Exercise 2.2 · Q6Simplest form of $\tan^{-1}\frac{x}{\sqrt{a^2-x^2}}$ worked
Exercise 2.2 · Q7Simplest form of $\tan^{-1}\frac{3a^2x-x^3}{a^3-3ax^2}$ worked
Exercise 2.2 · Q8 $\tan^{-1}\left[2\cos\left(2\sin^{-1}\frac12\right)\right]$ worked
Exercise 2.2 · Q9 $\tan\frac12\left[\sin^{-1}\frac{2x}{1+x^2}+\cos^{-1}\frac{1-y^2}{1+y^2}\right]$ worked
Exercise 2.2 · Q10 $\sin^{-1}\left(\sin\frac{2\pi}{3}\right)$ worked
Exercise 2.2 · Q11 $\tan^{-1}\left(\tan\frac{3\pi}{4}\right)$ worked
Exercise 2.2 · Q12 $\tan\left(\sin^{-1}\frac35+\cot^{-1}\frac32\right)$ worked
Exercise 2.2 · Q13 (MCQ) $\cos^{-1}\left(\cos\frac{7\pi}{6}\right)$ worked
Exercise 2.2 · Q14 (MCQ) $\sin\left(\frac{\pi}{3}-\sin^{-1}\left(-\frac12\right)\right)$ worked
Exercise 2.2 · Q15 (MCQ) $\tan^{-1}\sqrt3-\cot^{-1}\left(-\sqrt3\right)$ worked

Miscellaneous Exercise

Miscellaneous · Q1 $\cos^{-1}\left(\cos\frac{13\pi}{6}\right)$ worked
Miscellaneous · Q2 $\tan^{-1}\left(\tan\frac{7\pi}{6}\right)$ worked
Miscellaneous · Q3 $2\sin^{-1}\frac35=\tan^{-1}\frac{24}{7}$ worked
Miscellaneous · Q4 $\sin^{-1}\frac{8}{17}+\sin^{-1}\frac35=\tan^{-1}\frac{77}{36}$ worked
Miscellaneous · Q5 $\cos^{-1}\frac45+\cos^{-1}\frac{12}{13}=\cos^{-1}\frac{33}{65}$ worked
Miscellaneous · Q6 $\cos^{-1}\frac{12}{13}+\sin^{-1}\frac35=\sin^{-1}\frac{56}{65}$ worked
Miscellaneous · Q7 $\tan^{-1}\frac{63}{16}=\sin^{-1}\frac{5}{13}+\cos^{-1}\frac35$ worked
Miscellaneous · Q8 $\tan^{-1}\sqrt x=\frac12\cos^{-1}\frac{1-x}{1+x}$ worked
Miscellaneous · Q9A $\cot^{-1}$ of square roots equals $\frac x2$ worked
Miscellaneous · Q10A $\tan^{-1}$ of square roots equals $\frac\pi4-\frac12\cos^{-1}x$ worked
Miscellaneous · Q11Solve $2\tan^{-1}(\cos x)=\tan^{-1}(2\operatorname{cosec}x)$ worked
Miscellaneous · Q12Solve $\tan^{-1}\frac{1-x}{1+x}=\frac12\tan^{-1}x$ worked
Miscellaneous · Q13 (MCQ) $\sin(\tan^{-1}x)$ for $|x|<1$ worked
Miscellaneous · Q14 (MCQ)Solve $\sin^{-1}(1-x)-2\sin^{-1}x=\frac\pi2$ worked

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