Q 114. - Precalculus Enhanced with Graphing Utilities

Question

If $f\left(x\right)=\sin x ,g\left(x\right)=\cos x$ and $h\left(x\right)=2x ,P\left(x\right)=\frac{x}{2}$.Find the value of $\left(h·f\right) \left(\frac{5\pi }{6}\right)$.

Step-by-Step Solution

Answer

The value of $\left(h·f\right) \left(\frac{5\pi }{6}\right)$ is $\frac{5\pi }{6}$.

Step 1. Given information.

Given $h\left(x\right)=2x$ $,f\left(x\right)=\sin x$ and the value of $x$ is $\frac{5\pi }{6}$.

Step 2. Simplify the given expression.

$\left(h·f\right) \left(\frac{5\pi }{6}\right)$can be written as $h\left(\frac{5\pi }{6}\right)$ and $f\left(\frac{5\pi }{6}\right)$.

Substitute $2x$ for $h\left(\frac{5\pi }{6}\right)$ and $\sin x$for $f\left(\frac{5\pi }{6}\right)$.

$\left(h·f\right) \left(\frac{5\pi }{6}\right)=\left(2\times \frac{5\pi }{6} ·\sin \frac{5\pi }{6}\right)$

Step 3. Find the valuer of h · f   5 π 6 .

In trigonometric standard angle the value of $\sin \frac{5\pi }{6}$ is $\frac{1}{2}$.

Substitute the value in the simplified expression.

$$\begin{gathered} \left(h·f\right) \left(\frac{5\pi }{6}\right)=\left(2\times \frac{5\pi }{6} ·\sin \frac{5\pi }{6}\right) \\ =\left(\frac{5\pi }{3} ·\frac{1}{2}\right) \\ =\frac{5\pi }{6} \end{gathered}$$