Chapter 8.5, Problem 44AYU

In Problems 33-38, (a) use the Product-to-Sum Formulas to express each product as a sum, and (b) use the method of adding $y\text{-coordinates}$ to graph each function on the interval $\left[0,2\pi \right]$.

$h(x)=\cos (2x)\cos (x)$

To determine

To find: The expression of each product as sum and to graph the given function using the method of adding $y\text{-coordinates}$.

Answer to Problem 44AYU

$h\left(x\right) = \text{cos}2x\text{cos}x = \frac{1}{2}\left[\text{cos}x + \text{cos}3x\right]$

Explanation of Solution

Given:

$h\left(x\right) = \text{cos}2x\text{cos}x$

Formula used:

$\text{cos}A\text{cos}B = \frac{1}{2}\left[\text{cos}\left(A - B\right) + \text{cos}\left(A + B\right)\right]$

Calculation:

$h\left(x\right) = \text{cos}2x\text{cos}x$

$\text{cos}A\text{cos}B = \frac{1}{2}\left[\text{cos}\left(A - B\right) + \text{cos}\left(A + B\right)\right]$

$G\left(x\right) = \text{cos}2x\text{cos}x = \frac{1}{2}\left[\text{cos}\left(2x - x\right) + \text{cos}\left(2x + x\right)\right]$

$\text{cos}2x\text{cos}x = \frac{1}{2}\left[\text{cos}x + \text{cos}3x\right]$

Graph of $h\left(x\right) = \text{cos}2x\text{cos}x$

The graph for $h\left(x\right) = \text{cos}2x\text{cos}x$ = $\frac{1}{2}\left[\text{cos}x + \text{cos}3x\right]$ using the method of adding $y\text{-coordinates}$ is plotted above using the graphing calculator.