Q1. Find principal value for \(\sin^{-1}\left ( -\frac{1}{2} \right )\)
Soln:
Let \(\sin^{-1}\left ( -\frac{1}{2} \right )\) = a, then
\(\sin^{a} = -\frac{1}{2} = – \sin\frac{ \pi}{6} = \sin ( – \frac{ \pi}{6} )\)We know,
The principal value branch range for sin-1 is \(\left [ -\frac{ \pi}{2}, \frac{ \pi}{2} \right ]\) and \(\sin ( -\frac{ \pi}{6} ) = – \frac{1}{2}\)
Therefore principal value for \(\sin^{-1}\left ( -\frac{1}{2} \right ) \; is \; – \frac{ \pi}{6}\)
Q2. Find principal value for \(\cos^{-1}\left ( – \frac{\sqrt{3}}{2} \right )\)
Soln:
Let \(\cos^{-1}\left ( – \frac{\sqrt{3}}{2} \right )\) = a, then
\(\cos a = \frac{\sqrt{3}}{2} = \cos (\frac{\pi}{6})\)We know,
The principal value branch range for cos-1 is \(\left [ 0 , \pi \right ]\) and \(\cos (\frac{\pi}{6}) = \frac{\sqrt{3}}{2}\)
Therefore, principal value for \(\cos^{-1}\left ( – \frac{\sqrt{3}}{2} \right ) \; is \; \frac{\pi}{6}\)
Q3. Find principal value for cosec-1 (2)
Soln:
Let cosec-1 (2) = a. Then, cosec a = 2 = cosec \((\frac{\pi}{6} )\)
We know,
The principal value branch range for cosec-1 is \(\left [ -\frac{\pi}{2}, \frac{\pi}{2}\right ] – {0}\) and cosec\((\frac{\pi}{6} )\) = 2
Therefore, principal value for cosec‑1 (2) is \( \frac{\pi}{6}\)
Q4. Find principal value for \(\tan^{-1} \left ( – \sqrt{3} \right )\)
Soln:
Let \(\tan^{-1} \left ( – \sqrt{3} \right ) = a \)
Then, \(\tan = – \sqrt{3} = – \tan \frac{\pi}{3} \tan (- \frac{\pi}{3})\)
We know,
The principal value branch range for \(\tan^{-1} \; is \; \left [ -\frac{\pi}{2}, \frac{\pi}{2} \right ] \; and \; \tan\left ( -\frac{\pi}{3} \right ) = -\sqrt{3}\)
Therefore, principal value for \(\tan^{-1} \left ( – \sqrt{3} \right ) \; is \; -\frac{\pi}{3}\)
Q5. Find principal value for \(\cos^{-1}\left ( -\frac{1}{2} \right )\)
Soln:
Let \(\cos^{-1}\left ( -\frac{1}{2} \right )\) = a,
Then \(\cos a = -\frac{1}{2} = -cos \frac{\pi}{3} = \cos ( \pi – \frac{\pi}{3} ) = \cos( \frac{2 \pi}{3} )\)
We know,
The principal value branch range for \(\cos ^{-1} \; is \; \left [ 0 , \pi \right ] \; and \; \cos \left ( \frac{2 \pi}{3} \right ) = – \frac{1}{2}\)
Therefore, principal value for \(\cos^{-1}\left ( -\frac{1}{2} \right ) \; is \; \frac{2 \pi}{3}\)
Q6. Find principal value for \(\tan^{-1} (-1)\)
Soln:
Let \(\tan^{-1} (-1) = a\),
Then, tan a = -1 = \(-\tan ( \frac{\pi}{4} ) = \tan ( – \frac{\pi}{4} )\)
We know,
The principal value branch range for \(\tan^{-1} \; is \; \left ( -\frac{\pi}{2}, \frac{\pi}{2} \right ) \; and \; \tan ( – \frac{\pi}{4} ) = -1 \)
Therefore, principal value for \(\tan^{-1} (-1) \; is \; -\frac{\pi}{4}\)
Q7. Find principal value for \(\sec^{-1} \left ( \frac{2}{\sqrt{3}} \right )\)
Soln:
Let \(\sec^{-1} \left ( \frac{2}{\sqrt{3}} \right ) = a \),
Then \(\sec a = \frac{2}{\sqrt{3}} = \sec (\frac{\pi}{6})\)
We know,
The principal value branch range for \(\sec^{-1} \; is \; \left [ 0 , \pi \right ] – \left \{ \frac{ \pi }{2} \right \} \; and \; \sec (\frac{\pi}{6}) = \frac{2}{\sqrt{3}}\)
Therefore, principal value for \(\sec^{-1} \left ( \frac{2}{\sqrt{3}} \right ) \; is ; \frac{\pi}{6}\)
Q8. Find principal value for \(\cot^{-1} \sqrt{3}\)
Soln:
Let \(\cot^{-1} \sqrt{3} = a\),
Then \(\cot a = \sqrt {3} = \cot \left ( \frac{\pi}{6} \right )\)
We know,
The principal value branch range for cot-1 is \( ( 0 , \pi ) \) and \(\cot \left ( \frac{\pi}{6} \right ) = \sqrt{3}\)
Therefore, principal value for \(\cot^{-1} \sqrt{3} = \frac{\pi}{6}\)
Q9. Find principal value for \(\cos^{-1} \left ( – \frac{1}{\sqrt{2}} \right ) \)
Soln:
Let \(\cos^{-1} \left ( – \frac{1}{\sqrt{2}} \right ) = a \)
Then \(\cos a = \frac{-1}{\sqrt{2}} = – \cos \left ( \frac{\pi}{4} \right ) = \cos \left ( \pi – \frac{\pi}{4} \right ) = \cos \left ( \frac{3 \pi}{4} \right )\)
We know,
The principal value branch range for cos‑1 is \([0 , \pi] \; and \; \cos \left ( \frac{3 \pi}{4} \right ) = -\frac{1}{\sqrt{2}}\)
Therefore, principal value for \(\cos^{-1} \left ( – \frac{1}{\sqrt{2}} \right ) \; is \; \frac{3 \pi }{4}\)
Q10. Find principal value for cosec-1 \(\left ( -\sqrt{2} \right )\)
Soln:
Let cosec-1\(\left ( -\sqrt{2} \right )\) = a, Then
cosec a = \( -\sqrt{2} \) = -cosec\(\left ( \frac{\pi}{4}\right )\) = cosec \(\left ( -\frac{\pi}{4}\right )\)
We know,
The principal value branch range for cosec-1 is \(\left [ -\frac{\pi}{2} , \frac{\pi}{2} \right ] – \left \{ 0 \right \}\) and cosec\(\frac{-\pi}{4} = -\sqrt{2}\)
Therefore, principal value for cosec-1 \(\left ( -\sqrt{2} \right ) \; is \; -\frac{\pi}{4} \)
Q11. Solve \(\tan ^{-1}(1) + \cos^{-1} \left ( -\frac{1}{2} \right ) + \sin ^{-1}\left ( -\frac{1}{2} \right )\)
Soln:
Let \(\tan ^{-1}(1) = a \), then
\(\tan a = 1 = \tan \frac{\pi}{4}\)We know,
The principal value branch range for \(\tan ^{-1} \; is \; \left ( -\frac{\pi}{2}, \frac{\pi}{2} \right ) \)
\(\tan ^{-1}(1) = \frac{\pi}{4}\)Let \( \cos^{-1} \left ( -\frac{1}{2} \right ) = b \), then
\(\cos b = -\frac{1}{2} = -\cos \frac {\pi}{3} = \cos\left ( \pi – \frac{\pi}{3} \right ) = \cos\left ( \frac{2\pi}{3} \right )\)We know,
The principal value branch range for cos-1 is \([0 , \pi]\)
\(\cos ^{-1} \left ( -\frac{1}{2} \right ) = \frac{2 \pi }{3}\)Let \(\sin^{-1}\left ( -\frac{1}{2} \right ) = c\), then
\(\sin c = – \frac{1}{2} = – \sin \frac{\pi}{6} = \sin \left ( -\frac{\pi}{6} \right )\)We know,
The principal value branch range for \(\sin ^{-1} \; is \; \left [ -\frac{\pi}{2}, \frac{\pi}{2} \right ] \)
\(\sin^{-1} \left ( -\frac{1}{2} \right ) = – \frac{\pi}{6}\)Now
\(\tan ^{-1}(1) + \cos^{-1} \left ( -\frac{1}{2} \right ) + \sin ^{-1}\left ( -\frac{1}{2} \right )\) \(= \frac{\pi}{4} + \frac{2\pi}{3} – \frac{\pi}{6} = \frac{3\pi + 8\pi – 2\pi}{12} = \frac{9\pi}{12} = \frac{3\pi}{4}\)
Q12. Solve \(\cos ^{-1} \left ( \frac{1}{2} \right ) + 2 \sin ^{-1} \left ( \frac{1}{2} \right )\)
Soln:
Let \(\cos ^{-1} \left ( \frac{1}{2} \right ) = a \), then \(\cos a = \frac{1}{2} = \cos \frac{\pi}{3}\)
We know,
The principal value branch range for cos-1 is \(\left [0 , \pi \right ]\)
\(\cos ^{-1} \left ( \frac{1}{2} \right ) = \frac{\pi}{3}\)Let \(\sin ^{-1} \left (- \frac{1}{2} \right ) = b \), then \(\sin b = \frac{1}{2} = \sin \frac{\pi}{6}\)
We know,
The principal value branch range for \(\sin ^{-1} \; is \; \left [ -\frac{\pi}{2}, \frac{\pi}{2} \right ] \)
\(\sin ^{-1} \left ( \frac{1}{2} \right ) = \frac{\pi}{6}\)Now,
\(\cos ^{-1} \left ( \frac{1}{2} \right ) + 2 \sin ^{-1} \left ( \frac{1}{2} \right )\) \(= \frac{\pi}{3} + 2 \times \frac{\pi}{6} = \frac{\pi}{3} + \frac{\pi}{3} = \frac{2\pi}{3}\)
Q13. If sin-1 a = b, then
(i) \(0 \leq b \leq \pi\)
(ii) \(-\frac{\pi}{2} \leq b \leq \frac{\pi}{2}\)
(iii) \(0 < b < \pi\)
(iv) \(-\frac{\pi}{2} < b < \frac{\pi}{2}\)
Soln:
Given sin-1 a = b
We know,
The principal value branch range for \(\sin ^{-1} \; is \; \left [ -\frac{\pi}{2}, \frac{\pi}{2} \right ] \)
Therefore, \(-\frac{\pi}{2} \leq b \leq \frac{\pi}{2}\)
Q14. The value of \(\tan ^{-1} \sqrt{3} – \sec ^{-1}(-2)\) is
(i) \( \pi\)
(ii) \( – \frac{\pi}{3}\)
(iii) \(\frac{\pi}{3}\)
(iv) \(\frac{2 \pi}{3}\)
Soln:
Let \(\tan ^{-1} \sqrt{3} = a \), then
\(\tan a = \sqrt{3} = \tan \frac{\pi}{3}\)We know
The principal value branch range for \(\tan ^{-1} \; is \; \left ( -\frac{\pi}{2}, \frac{\pi}{2} \right ) \)
\(\tan ^{-1}\sqrt{3} = \frac{\pi}{3}\)Let sec-1(-2) = b, then
sec b = -2 = \(– \sec \frac{\pi}{3} = \sec \left ( \pi – \frac{\pi}{3} \right ) = \sec \left ( \frac{2 \pi}{3} \right )\)
We know
The principal value branch range for sec-1 is \([0 , \pi] – \left \{ \frac{\pi}{2} \right \}\)
\(\sec ^{-1}(-2) = \frac{2 \pi}{3}\)Now,
\(\tan ^{-1} \sqrt{3} – \sec ^{-1}(-2) = \frac{\pi}{3} – \frac{2 \pi}{3} = – \frac{\pi}{3}\)Hence option (ii) is correct
