Question

In Problems 67–78, $f\left(x\right)=\sin x,-\frac{\mathrm{\pi }}{2}\le x\le \frac{\mathrm{\pi }}{2},g\left(x\right)=\cos x,0\le x\le \mathrm{\pi }\mathrm{and}\mathrm{h}\left(\mathrm{x}\right)=\mathrm{tanx},-\frac{\mathrm{\pi }}{2}<x<\frac{\mathrm{\pi }}{2}$

Find the exact value of each composite function.

role="math" localid="1646624335863" $g^{-1}\left(f\left(-\frac{5\mathrm{\pi }}{6}\right)\right)$

Short answer

The required exact value of given expression is $\frac{2\mathrm{\pi }}{3}$.

Step-by-step solution

Step 1. Given information. 

The composite function$g^{-1}\left(f\left(-\frac{5\mathrm{\pi }}{6}\right)\right)$.

Step 2. Calculate f-5π6.

$$\begin{gathered} f\left(-\frac{5\mathrm{\pi }}{6}\right)=\sin \left(-\frac{5\mathrm{\pi }}{6}\right) \\ =-\frac{1}{2} \end{gathered}$$

Step 3. Now, find g-1f-5π6.

$$\begin{gathered} g^{-1}\left(f\left(-\frac{5\mathrm{\pi }}{6}\right)\right)=g^{-1}\left(-\frac{1}{2}\right) \\ Let, \\ {g}^{-1}\left(-\frac{1}{2}\right)=x \\ g\left(x\right)=\frac{-1}{2} \\ \cos x=\frac{-1}{2} \\ \sin ce,\cos x=\frac{-1}{2}forx=\frac{2\mathrm{\pi }}{3}ontheinterval0\le x\le \mathrm{\pi }. \\ \mathrm{So}, \\ {\mathrm{g}}^{-1}\left(\mathrm{f}\left(-\frac{5\mathrm{\pi }}{6}\right)\right)=\mathrm{x}=\frac{2\mathrm{\pi }}{3} \end{gathered}$$