Question

In problem, $f\left(x\right)=\sin x,-\frac{\pi }{2}\le x\le \frac{\pi }{2},g\left(x\right)=\cos x,0\le x\le \pi$and $h\left(x\right)=\tan x,-\frac{\pi }{2}\le x\le \frac{\pi }{2}$. Find the value of the composite function.

$\begin{array}{rcl}& & f^{-1}\left(g\left(\frac{5\mathrm{\pi }}{6}\right)\right)\end{array}$

Short answer

Value of the composite function is$-\frac{\mathrm{\pi }}{3}$.

Step-by-step solution

Step 1. Given information

$f\left(x\right)=\sin x,-\frac{\pi }{2}\le x\le \frac{\pi }{2},g\left(x\right)=\cos x,0\le x\le \pi$and $h\left(x\right)=\tan x,-\frac{\pi }{2}\le x\le \frac{\pi }{2}$

Step 2. Value of the composite function.

It is given that $g\left(x\right)=\cos x$.

width="157">g5π6=cos5π6=cosπ-π6=-cosπ6=-32

$\begin{array}{rcl}{f}^{-1}\left(g\left(\frac{5\mathrm{\pi }}{6}\right)\right)& =& f^{-1}\left(-\frac{\sqrt{3}}{2}\right)\\ & =& {\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)\end{array}$

Let .

$\begin{array}{rcl}\sin y& =& -\frac{\sqrt{3}}{2}\\ & =& -\sin \left(\frac{\mathrm{\pi }}{3}\right)\\ & =& \sin \left(-\frac{\mathrm{\pi }}{3}\right)\end{array}$

So,

localid="1647840234802" $\begin{array}{rcl}{f}^{-1}\left(g\left(\frac{5\mathrm{\pi }}{6}\right)\right)& =& -\frac{\mathrm{\pi }}{3}\end{array}$