Chapter 7.3, Problem 108E

Touch-Tone Telephones When a key is pressed on a touchtone telephone, the keypad generates two pure tones, which combine to produce a sound that uniquely identifies the key. The figure shows the low frequency f₁ and the high frequency f₂ associated with each key. Pressing a key produces the sound wave y = sin(2πf₁t) + sin(2πf₂t).

1. (a) Find the function that models the sound produced when the 4 key is pressed.
2. (b) Use a Sum-to-Product Formula to express the sound generated by the 4 key as a product of a sine and a cosine function.
3. (c) Graph the sound wave generated by the 4 key from t = 0 to t = 0.006 s.

To determine

To find: The function that models the sound produced when 4 key is pressed.

Answer to Problem 108E

The function that models the sound produced when 4 key is pressed is $y=\sin \left(1540\pi t\right)+\sin \left(2418\pi t\right)$.

Explanation of Solution

Given:

Pressed key produces the sound wave $y=\sin \left(2\pi f_{1}t\right)+\sin \left(2\pi f_{2}t\right)$.

Calculation:

When the 4 key is pressed, it produces a low frequency sound $f_1=770$ Hz and a high frequency sound $f_2=1209$ Hz.

The sound wave produced is $y=\sin \left(2\pi f_{1}t\right)+\sin \left(2\pi f_{2}t\right)$.

Substitute the value of $f_1=770$ and $f_2=1209$ in $y=\sin \left(2\pi f_{1}t\right)+\sin \left(2\pi f_{2}t\right)$.

$\begin{array}{c}y=\sin \left(2\pi f_{1}t\right)+\sin \left(2\pi f_{2}t\right)\\ =\sin \left(2\pi 770t\right)+\sin \left(2\pi 1209t\right)\\ =\sin \left(1540\pi t\right)+\sin \left(2418\pi t\right)\end{array}$

Hence, the function that models the sound produced when 4 key is pressed is $y=\sin \left(1540\pi t\right)+\sin \left(2418\pi t\right)$.

To determine

To express: The sound generated by the 4 key as a product of a sine and a cosine function.

Answer to Problem 108E

The sound generated by the 4 key as a product of a sine and a cosine function is $y=2\sin \left(1979\pi t\right)\cos \left(439\pi t\right)$.

Explanation of Solution

Formula used:

Sum-To-Product formulae:

$\begin{array}{l}\sin x+\sin y=2\sin \frac{x+y}{2}\cos \frac{x-y}{2}\\ \sin x-\sin y=2\cos \frac{x+y}{2}\sin \frac{x-y}{2}\\ \cos x+\cos y=2\cos \frac{x+y}{2}\cos \frac{x-y}{2}\\ \cos x-\cos y=-2\sin \frac{x+y}{2}\sin \frac{x-y}{2}\end{array}$

Calculation:

Since $y=\sin \left(1540\pi t\right)+\sin \left(2418\pi t\right)$, substitute this formula in $\sin x+\sin y=2\sin \frac{x+y}{2}\cos \frac{x-y}{2}$.

$\begin{array}{c}y=\sin \left(1540\pi t\right)+\sin \left(2418\pi t\right)\\ =2\sin \left(\frac{1540\pi t+2418\pi t}{2}\right)\cos \left(\frac{1540\pi t-2418\pi t}{2}\right)\\ =2\sin \left(\frac{3958\pi t}{2}\right)\cos \left(\frac{-878\pi t}{2}\right)\\ =2\sin \left(1979\pi t\right)\cos \left(-439\pi t\right)\end{array}$

Simplify further.

$\begin{array}{c}y=2\sin \left(1979\pi t\right)\cos \left(-439\pi t\right)\\ =2\sin \left(1979\pi t\right)\cos \left(439\pi t\right)\left(\because \cos \left(-x\right)=\cos x\right)\end{array}$

Hence, the sound generated by the 4 key as a product of a sine and a cosine function $y=2\sin \left(1979\pi t\right)\cos \left(439\pi t\right)$.

To determine

To sketch: The sound wave generated by the key 4 from $t=0$ to $t=0.006$ s.

Answer to Problem 108E

Use online graphing calculator to sketch the graph of the functions $y=2\sin \left(1979\pi t\right)\cos \left(439\pi t\right)$ between  $t=0$ to $t=0.006$ s as shown below in Figure 1.

Explanation of Solution

Calculation:

Use online graphing calculator to sketch the graph of the functions $y=2\sin \left(1979\pi t\right)\cos \left(439\pi t\right)$ between $t=0$ to $t=0.006$ s as shown below in Figure 1

From Figure 1, it is observed that the sound wave generated by the key 4 from $t=0$ to $t=0.006$ s.