Mathematics · Textbook solutions

Inverse Trigonometric Functions

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 49 questions

2.2 Basic Concepts

16 q

Solved Examples

Worked · 2
  1. 2.1 Eg.1
    Find the principal value of sin⁡−1(12)\sin^{-1}\left(\frac{1}{\sqrt{2}}\right).
  2. 2.1 Eg.2
    Find the principal value of cot⁡−1(−13)\cot^{-1}\left(\frac{-1}{\sqrt{3}}\right).

Exercise 2.1

Practice · 14
  1. Ex 2.1 Q1
    Find the principal value of sin⁡−1(−12)\sin^{-1}\left(-\frac{1}{2}\right).
  2. Ex 2.1 Q2
    Find the principal value of cos⁡−1(32)\cos^{-1}\left(\frac{\sqrt{3}}{2}\right).
  3. Ex 2.1 Q3
    Find the principal value of cosec⁡−1(2)\operatorname{cosec}^{-1}(2).
  4. Ex 2.1 Q4
    Find the principal value of tan⁡−1(−3)\tan^{-1}\left(-\sqrt{3}\right).
  5. Ex 2.1 Q5
    Find the principal value of cos⁡−1(−12)\cos^{-1}\left(-\frac{1}{2}\right).
  6. Ex 2.1 Q6
    Find the principal value of tan⁡−1(−1)\tan^{-1}(-1).
  7. Ex 2.1 Q7
    Find the principal value of sec⁡−1(23)\sec^{-1}\left(\frac{2}{\sqrt{3}}\right).
  8. Ex 2.1 Q8
    Find the principal value of cot⁡−1(3)\cot^{-1}\left(\sqrt{3}\right).
  9. Ex 2.1 Q9
    Find the principal value of cos⁡−1(−12)\cos^{-1}\left(-\frac{1}{\sqrt{2}}\right).
  10. Ex 2.1 Q10
    Find the principal value of cosec⁡−1(−2)\operatorname{cosec}^{-1}\left(-\sqrt{2}\right).
  11. Ex 2.1 Q11
    Find the value of tan⁡−1(1)+cos⁡−1(−12)+sin⁡−1(−12)\tan^{-1}(1) + \cos^{-1}\left(-\frac{1}{2}\right) + \sin^{-1}\left(-\frac{1}{2}\right).
  12. Ex 2.1 Q12
    Find the value of cos⁡−1(12)+2sin⁡−1(12)\cos^{-1}\left(\frac{1}{2}\right) + 2\sin^{-1}\left(\frac{1}{2}\right).
  13. Ex 2.1 Q13
    If sin⁡−1x=y\sin^{-1} x = y, then
    1. A.
      0≤y≤π0 \le y \le \pi
    2. B.
      −π2≤y≤π2-\frac{\pi}{2} \le y \le \frac{\pi}{2}
    3. C.
      0<y<π0 < y < \pi
    4. D.
      −π2<y<π2-\frac{\pi}{2} < y < \frac{\pi}{2}
  14. Ex 2.1 Q14
    tan⁡−13−sec⁡−1(−2)\tan^{-1}\sqrt{3} - \sec^{-1}(-2) is equal to
    1. A.
      π\pi
    2. B.
      −π3-\frac{\pi}{3}
    3. C.
      π3\frac{\pi}{3}
    4. D.
      2π3\frac{2\pi}{3}

2.3 Properties of Inverse Trigonometric Functions

18 q

Solved Examples

Worked · 3
  1. 2.2 Eg.3
    Show that (i) sin⁡−1(2x1−x2)=2sin⁡−1x, −12≤x≤12\sin^{-1}\left(2x\sqrt{1-x^2}\right) = 2\sin^{-1} x,\ -\frac{1}{\sqrt{2}} \le x \le \frac{1}{\sqrt{2}} (ii) sin⁡−1(2x1−x2)=2cos⁡−1x, 12≤x≤1\sin^{-1}\left(2x\sqrt{1-x^2}\right) = 2\cos^{-1} x,\ \frac{1}{\sqrt{2}} \le x \le 1
  2. 2.2 Eg.4
    Express tan⁡−1(cos⁡x1−sin⁡x), −3π2<x<π2\tan^{-1}\left(\frac{\cos x}{1-\sin x}\right),\ -\frac{3\pi}{2} < x < \frac{\pi}{2} in the simplest form.
  3. 2.2 Eg.5
    Write cot⁡−1(1x2−1), x>1\cot^{-1}\left(\frac{1}{\sqrt{x^2-1}}\right),\ x > 1 in the simplest form.

Exercise 2.2

Practice · 15
  1. Ex 2.2 Q1
    Prove that 3sin⁡−1x=sin⁡−1(3x−4x3), x∈[−12,12]3\sin^{-1} x = \sin^{-1}\left(3x - 4x^3\right),\ x \in \left[-\frac{1}{2}, \frac{1}{2}\right].
  2. Ex 2.2 Q2
    Prove that 3cos⁡−1x=cos⁡−1(4x3−3x), x∈[12,1]3\cos^{-1} x = \cos^{-1}\left(4x^3 - 3x\right),\ x \in \left[\frac{1}{2}, 1\right].
  3. Ex 2.2 Q3
    Write the following function in the simplest form: tan⁡−1(1+x2−1x), x≠0\tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right),\ x \ne 0.
  4. Ex 2.2 Q4
    Write the following function in the simplest form: tan⁡−1(1−cos⁡x1+cos⁡x), 0<x<π\tan^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right),\ 0 < x < \pi.
  5. Ex 2.2 Q5
    Write the following function in the simplest form: tan⁡−1(cos⁡x−sin⁡xcos⁡x+sin⁡x), −π4<x<3π4\tan^{-1}\left(\frac{\cos x - \sin x}{\cos x + \sin x}\right),\ \frac{-\pi}{4} < x < \frac{3\pi}{4}.
  6. Ex 2.2 Q6
    Write the following function in the simplest form: tan⁡−1(xa2−x2), ∣x∣<a\tan^{-1}\left(\frac{x}{\sqrt{a^2-x^2}}\right),\ |x| < a.
  7. Ex 2.2 Q7
    Write the following function in the simplest form: tan⁡−1(3a2x−x3a3−3ax2), a>0; −a3<x<a3\tan^{-1}\left(\frac{3a^2 x - x^3}{a^3 - 3a x^2}\right),\ a > 0;\ \frac{-a}{\sqrt{3}} < x < \frac{a}{\sqrt{3}}.
  8. Ex 2.2 Q8
    Find the value of tan⁡−1[2cos⁡(2sin⁡−112)]\tan^{-1}\left[2\cos\left(2\sin^{-1}\frac{1}{2}\right)\right].
  9. Ex 2.2 Q9
    Find the value of tan⁡12[sin⁡−12x1+x2+cos⁡−11−y21+y2], ∣x∣<1, y>0\tan\frac{1}{2}\left[\sin^{-1}\frac{2x}{1+x^2} + \cos^{-1}\frac{1-y^2}{1+y^2}\right],\ |x| < 1,\ y > 0 and xy<1xy < 1.
  10. Ex 2.2 Q10
    Find the value of sin⁡−1(sin⁡2π3)\sin^{-1}\left(\sin\frac{2\pi}{3}\right).
  11. Ex 2.2 Q11
    Find the value of tan⁡−1(tan⁡3π4)\tan^{-1}\left(\tan\frac{3\pi}{4}\right).
  12. Ex 2.2 Q12
    Find the value of tan⁡(sin⁡−135+cot⁡−132)\tan\left(\sin^{-1}\frac{3}{5} + \cot^{-1}\frac{3}{2}\right).
  13. Ex 2.2 Q13
    cos⁡−1(cos⁡7π6)\cos^{-1}\left(\cos\frac{7\pi}{6}\right) is equal to
    1. A.
      7π6\frac{7\pi}{6}
    2. B.
      5π6\frac{5\pi}{6}
    3. C.
      π3\frac{\pi}{3}
    4. D.
      π6\frac{\pi}{6}
  14. Ex 2.2 Q14
    sin⁡(π3−sin⁡−1(−12))\sin\left(\frac{\pi}{3} - \sin^{-1}\left(-\frac{1}{2}\right)\right) is equal to
    1. A.
      12\frac{1}{2}
    2. B.
      13\frac{1}{3}
    3. C.
      14\frac{1}{4}
    4. D.
      11
  15. Ex 2.2 Q15
    tan⁡−13−cot⁡−1(−3)\tan^{-1}\sqrt{3} - \cot^{-1}\left(-\sqrt{3}\right) is equal to
    1. A.
      π\pi
    2. B.
      −π2-\frac{\pi}{2}
    3. C.
      00
    4. D.
      232\sqrt{3}

Miscellaneous Exercise on Chapter 2

15 q

Solved Examples

Worked · 1
  1. Misc Eg.6
    Find the value of sin⁡−1(sin⁡3π5)\sin^{-1}\left(\sin\frac{3\pi}{5}\right).

Miscellaneous Exercise

Practice · 14
  1. Misc Q1
    Find the value of cos⁡−1(cos⁡13π6)\cos^{-1}\left(\cos\frac{13\pi}{6}\right).
  2. Misc Q2
    Find the value of tan⁡−1(tan⁡7π6)\tan^{-1}\left(\tan\frac{7\pi}{6}\right).
  3. Misc Q3
    Prove that 2sin⁡−135=tan⁡−12472\sin^{-1}\frac{3}{5} = \tan^{-1}\frac{24}{7}.
  4. Misc Q4
    Prove that sin⁡−1817+sin⁡−135=tan⁡−17736\sin^{-1}\frac{8}{17} + \sin^{-1}\frac{3}{5} = \tan^{-1}\frac{77}{36}.
  5. Misc Q5
    Prove that cos⁡−145+cos⁡−11213=cos⁡−13365\cos^{-1}\frac{4}{5} + \cos^{-1}\frac{12}{13} = \cos^{-1}\frac{33}{65}.
  6. Misc Q6
    Prove that cos⁡−11213+sin⁡−135=sin⁡−15665\cos^{-1}\frac{12}{13} + \sin^{-1}\frac{3}{5} = \sin^{-1}\frac{56}{65}.
  7. Misc Q7
    Prove that tan⁡−16316=sin⁡−1513+cos⁡−135\tan^{-1}\frac{63}{16} = \sin^{-1}\frac{5}{13} + \cos^{-1}\frac{3}{5}.
  8. Misc Q8
    Prove that tan⁡−1x=12cos⁡−1(1−x1+x), x∈[0,1]\tan^{-1}\sqrt{x} = \frac{1}{2}\cos^{-1}\left(\frac{1-x}{1+x}\right),\ x \in [0, 1].
  9. Misc Q9
    Prove that cot⁡−1(1+sin⁡x+1−sin⁡x1+sin⁡x−1−sin⁡x)=x2, x∈(0,π4)\cot^{-1}\left(\frac{\sqrt{1+\sin x} + \sqrt{1-\sin x}}{\sqrt{1+\sin x} - \sqrt{1-\sin x}}\right) = \frac{x}{2},\ x \in \left(0, \frac{\pi}{4}\right).
  10. Misc Q10
    Prove that tan⁡−1(1+x−1−x1+x+1−x)=π4−12cos⁡−1x, −12≤x≤1\tan^{-1}\left(\frac{\sqrt{1+x} - \sqrt{1-x}}{\sqrt{1+x} + \sqrt{1-x}}\right) = \frac{\pi}{4} - \frac{1}{2}\cos^{-1} x,\ -\frac{1}{\sqrt{2}} \le x \le 1. [Hint: Put x=cos⁡2θx = \cos 2\theta]
  11. Misc Q11
    Solve the equation 2tan⁡−1(cos⁡x)=tan⁡−1(2cosec⁡x)2\tan^{-1}(\cos x) = \tan^{-1}(2\operatorname{cosec} x).
  12. Misc Q12
    Solve the equation tan⁡−11−x1+x=12tan⁡−1x, (x>0)\tan^{-1}\frac{1-x}{1+x} = \frac{1}{2}\tan^{-1} x,\ (x > 0).
  13. Misc Q13
    sin⁡(tan⁡−1x), ∣x∣<1\sin\left(\tan^{-1} x\right),\ |x| < 1 is equal to
    1. A.
      x1−x2\frac{x}{\sqrt{1-x^2}}
    2. B.
      11−x2\frac{1}{\sqrt{1-x^2}}
    3. C.
      11+x2\frac{1}{\sqrt{1+x^2}}
    4. D.
      x1+x2\frac{x}{\sqrt{1+x^2}}
  14. Misc Q14
    sin⁡−1(1−x)−2sin⁡−1x=π2\sin^{-1}(1-x) - 2\sin^{-1} x = \frac{\pi}{2}, then xx is equal to
    1. A.
      0, 120,\ \frac{1}{2}
    2. B.
      1, 121,\ \frac{1}{2}
    3. C.
      00
    4. D.
      12\frac{1}{2}