Question

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

Answer 1

We know that principal value branch of

${cosec}^{ - 1} x \:\: is\:\: [-\frac{\pi}{2}, \frac{\pi}{2}]$-{0}

To solve the given equation we must keep consider that cosec and cosec inverse cancels each other only if x belongs to its principal value .
$\sqrt{3} cosec\: x + 2=0 \\ \\\sqrt{3} cosec\: x = - 2 \\ \\ cosec\: x = \frac{ - 2}{ \sqrt{3} } \\ \\ x = {cosec}^{ - 1} ( \frac{ - 2}{ \sqrt{3} } )\\ \\ we \: know \: that \: cosec \: \frac{\pi}{3} = \frac{2}{ \sqrt{3} } \\ \\We\:\: know\:\:that\:\: \\ \\ {cosec}^{ - 1} ( - cosec\: x) = - {cosec}^{ - 1} ( cosec\: x) \\\\ So\\\\ x= {cosec}^{ - 1} (cosec ( -\frac{\pi}{3} )) \\ \\x=- {cosec}^{ - 1} (cosec ( \frac{\pi}{3} ))\\\\ x= - \frac{ \pi}{3} \: \: \: \: (x \: belongs \: to \: [- \frac{\pi}{2}, \frac{\pi}{2}])-(0)\\ \\ x = \frac{ - \pi}{3}$

is the solution of the equation.