Question

The value of \(\arctan \left( {1/\sqrt 3 } \right) + \operatorname{arcsec} \left( { - 2} \right)\) is:
1. \(\frac{\pi }{2}\)
2. \( \frac {5\pi}{6} \)
3. \(- \frac{\pi }{2}\)
4. 0

Answer 1

Correct Answer - Option 2 : \( \frac {5\pi}{6} \)

Concept:

\(\arctan x = y ⇒ x = \tan y\)

Calculation:

Here, we have to find the value of \(\arctan \left( {1/\sqrt 3 } \right) + \operatorname{arcsec} \left( { - 2} \right)\)

\(\tan x = \frac{1}{{\sqrt 3 }}\) ⇒ \(x = \frac{\pi }{6}\)

Also, \(\sec y = -2\) ⇒ \(y = \frac{2\pi }{3}\)

So, \(\arctan \left( {1/\sqrt 3 } \right) + {\mathop{\rm arcsec}\nolimits} \left( { - 2} \right) = x + y\)

\(= \frac{\pi }{6} + \frac{2\pi }{3}\)

\(= \frac{\pi+4\pi}{6} = \frac {5\pi}{6} \)