Question

(a) Use the Product-to-Sum Formulas to express each product as a sum, and

(b) Use the method of adding y-coordinates to graph each function on the interval $\left[0,2\mathrm{\pi }\right]$

role="math" localid="1646907271880" $F\left(x\right)=\sin \left(3x\right)\sin x$

Short answer

$\left(a\right)F\left(x\right)=\frac{1}{2}\left[\cos \left(2x\right)-\cos \left(4x\right)\right]$

(b)

Step-by-step solution

Step 1. Given Information

The given function is$F\left(x\right)=\sin \left(3x\right)\sin x$

Part(a) Step 1. Calculation

First multiply and divide the given function by 2 and then use the product to sum formula to obtain the required result.

$$\begin{gathered} F\left(x\right)=\frac{1}{2}\left(\cos \right(3x-x)-\cos (3x+x\left)\right) \\ =\frac{1}{2}\left(\cos \right(2x)-\cos (4x\left)\right) \end{gathered}$$

Part(b) Step 1. Explanation and Graph 

Graph the component functions obtained in above part in same coordinate system and then select several values of x at which we can compute the value of y.

Plot the points and connect them to get the required graph.