Question

Find the principal solution of the equation.
√3 sec x + 2= 0

Answer 1

Solution:

As we know that principal value branch of sec inverse is [0,π]-{ π/2}

thus while solving the equation we must remember that sec inverse cancel sec x only if x belongs to principal value branch.

$sec\:x=\frac{-2}{\sqrt{3}}\\ \\ x= {sec}^{-1}[\frac{-2}{\sqrt{3}}] \\ \\ x = {sec}^{ - 1} (-sec (π/6))$

we know that ${sec}^{ - 1}(-x) = \pi -{sec}^{ - 1} x$
So,

$x = {sec}^{ - 1} (-sec (π/6))\\\\x =\pi- {sec}^{ - 1} (sec (π/6))$

here both cancels with each other because π/6 belongs to [0,π]-{ π/2}

Thus

$x = π-π/6\\\\ x = 5π/6$