Chapter 7.3, Problem 107E

(a)

To determine

To sketch: The function $y=f\left(t\right)$.

Answer to Problem 107E

Use online graphing calculator to sketch the graph of the function $y=f\left(t\right)$ as shown below in Figure 1:

Explanation of Solution

Given:

Consider the function $f\left(t\right)=f_1\left(t\right)+f_2\left(t\right)$, where $f_1\left(t\right)=\cos 11t$, $f_2\left(t\right)=\cos 13t$.

Calculation:

Use online graphing calculator to sketch the graph of the function $y=f\left(t\right)$ as shown below in Figure 1:

From Figure 1, it can be observed that the graph of the function $y=f\left(t\right)=\cos 11t+\cos 13t$.

(b)

To determine

To verify: The equation $f\left(t\right)=2\cos t\cos 12t$.

Explanation of Solution

Formula used:

Sum-To-Product formulae:

Sum-To-Product formulae: $\cos x+\cos y=2\cos \frac{x+y}{2}\cos \frac{x-y}{2}$.

Proof:

Substitute $f_1\left(t\right)=\cos 11t$ and $f_2\left(t\right)=\cos 13t$ in the function $f\left(t\right)=f_1\left(t\right)+f_2\left(t\right)$, and further simplify with the formula $\cos x+\cos y=2\cos \frac{x+y}{2}\cos \frac{x-y}{2}$.

$\begin{array}{c}f\left(t\right)=f_1\left(t\right)+f_2\left(t\right)\\ =\cos 11t+\cos 13t\\ =2\cos \left(\frac{11t+13t}{2}\right)\cos \left(\frac{11t-13t}{2}\right)\\ =2\cos \left(\frac{24t}{2}\right)\cos \left(\frac{-2t}{2}\right)\end{array}$

Further simplify,

$2\cos \left(\frac{24t}{2}\right)\cos \left(\frac{-2t}{2}\right)=2\cos 12t\cos t\left(\because \cot \left(-t\right)=\cos t\right)$.

Hence, the equation $f\left(t\right)=2\cos t\cos 12t$ is verified.

(c)

To determine

To sketch: The $y=2\cos t$, $y=-2\cos t$ and $y=f\left(t\right)=\cos 11t+\cos 13t$ together and describe their variation in the loudness of the sound.

Answer to Problem 107E

Use online graphing calculator to sketch the graph of the functions $y=2\cos t$, $y=-2\cos t$ and $y=f\left(t\right)$ as shown below in Figure 2.

The loudness of the sound varies between $y=-2\cos t$ to $y=2\cos t$.

Explanation of Solution

Calculation:

Use online graphing calculator to sketch the graph of the functions $y=2\cos t$, $y=-2\cos t$ and $y=f\left(t\right)$ as shown below in Figure 2.

From Figure 2, it is observed that the loudness of the sound varies between $y=-2\cos t$ to $y=2\cos t$.