Q 110. - 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 $\left(f·g\right) \left(\frac{4\pi }{3}\right)$.

Step-by-Step Solution

Answer

The value of $\left(f·g\right) \left(\frac{4\pi }{3}\right)$ is $\frac{\sqrt{3}}{4}$.

Step 1. Given information.

Given $f\left(x\right)=\sin x ,g\left(x\right)=\cos x$ and $h=2x ,P\left(x\right)=\frac{x}{2}$and the value of $x$ is $\frac{4\pi }{3}$.

Step 2. Simplify the given expression.

$\left(f·g\right) \left(\frac{4\pi }{3}\right)$ can be written as $f\left(\frac{4\pi }{3}\right)$ and $g\left(\frac{4\pi }{3}\right)$.

Substitute $\sin \left(\frac{4\pi }{3}\right)$ for $f\left(\frac{4\pi }{3}\right)$ and $\cos \left(\frac{4\pi }{3}\right)$ for $g\left(\frac{4\pi }{3}\right)$.

then,

$\left(f·g\right) \left(\frac{4\pi }{3}\right)=\left(\sin \frac{4\pi }{3}·\cos \frac{4\pi }{3}\right)$

Step 3. Find the value of f · g   4 π 3 .

In trigonometric standard angle the value of  $\sin \left(\frac{4\pi }{3}\right)$ is $\frac{-\sqrt{3}}{2}$ and the value of $\cos \left(\frac{4\pi }{3}\right)$ is $\frac{-1}{2}$.

Substitute the values of angles in the simplified expression.

$$\begin{gathered} \left(f·g\right) \left(\frac{4\pi }{3}\right)=\left(\sin \frac{4\pi }{3} ·\cos \frac{4\pi }{3}\right) \\ =\left(\frac{-\sqrt{3}}{2}·\frac{-1}{2}\right) \\ =\frac{\sqrt{3}}{4} \end{gathered}$$