Q 109. - 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 $\left(f·g\right) \left(\frac{3\pi }{4}\right)$.

Step-by-Step Solution

Answer

The value of $\left(f·g\right) \left(\frac{3\pi }{4}\right)$ is $-\frac{1}{2}$.

Step 1. Given information.

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

Step 2. Simplify the given expression.

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

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

then,

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

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

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

Substitute the values of angles in simplified expression.

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