Chapter 7.7, Problem 47AYU

a.

To determine

Write the sound as a product of sine and cosine.

Answer to Problem 47AYU

  $\boxed{y=2\sin \left(2061\pi t\right)\cos \left(357\pi t\right)}$

Explanation of Solution

Given information:

If $\tan \theta =a\tan \frac{\theta }{3}$ , express $\tan \frac{\theta }{3}$ in terms of $a$ .

On a touch −tone phone, each button produces a unique sound. The sound produced is the sum of two tones, given by

  $y=\sin \left(2\pi /t\right)$ and $y=\sin \left(2\pi ht\right)$

Where $l$ and $h$ are the low and high frequencies (cycles per second) shown in the illustration. For example, if you touch $7$ , the low frequency is $l=852$ cycles per second and the high frequency is $h=1209$ cycles per second. The sound emitted by touching $7$ is

  $y=\sin \left[2\pi \left(852\right)t\right]+\sin \left[2\pi \left(1209\right)t\right]$

  

Write this sound as a product of sine and cosine.

Calculation:

Consider the given function,

  $y=\sin \left[2\pi \left(852\right)t\right]+\sin \left[2\pi \left(1209\right)t\right]$

Apply sum to product formula, we get

  $\sin \alpha +\sin \beta =2\sin \left(\frac{\alpha +\beta }{2}\right)\cos \left(\frac{\alpha -\beta }{2}\right)$

  $\begin{array}{l}y=\sin \left[2\pi \left(852\right)t\right]+\sin \left[2\pi \left(1209\right)t\right]\\ =2\sin \left(\frac{2\pi \left(852\right)t+2\pi \left(1209\right)t}{2}\right)\cos \left(\frac{2\pi \left(852\right)t-2\pi \left(1209\right)t}{2}\right)\\ =\sin \left(2\pi \left(852\right)t\right)+\sin \left(2\pi \left(1209\right)t\right)=2\sin \left(2061\pi t\right)\cos \left(357\pi t\right)\\ \boxed{y=2\sin \left(2061\pi t\right)\cos \left(357\pi t\right)}\end{array}$

Hence, $\boxed{y=2\sin \left(2061\pi t\right)\cos \left(357\pi t\right)}$ .

b.

To determine

Find the maximum value of $y$ .

Answer to Problem 47AYU

  $\boxed{y=2}$

Explanation of Solution

Given information:

If $\tan \theta =a\tan \frac{\theta }{3}$ , express $\tan \frac{\theta }{3}$ in terms of $a$ .

On a touch −tone phone, each button produces a unique sound. The sound produced is the sum of two tones, given by

  $y=\sin \left(2\pi /t\right)$ and $y=\sin \left(2\pi ht\right)$

Where $l$ and $h$ are the low and high frequencies (cycles per second) shown in the illustration. For example, if you touch $7$ , the low frequency is $l=852$ cycles per second and the high frequency is $h=1209$ cycles per second. The sound emitted by touching $7$ is

  $y=\sin \left[2\pi \left(852\right)t\right]+\sin \left[2\pi \left(1209\right)t\right]$

  

Find the maximum value of $y$ .

Calculation:

Consider the given function,

  $y=\sin \left[2\pi \left(852\right)t\right]+\sin \left[2\pi \left(1209\right)t\right]$

The maximum value of $y$ .

  $\boxed{y=2}$

Hence, $\boxed{y=2}$ .

c.

To determine

Graph the sound emitted by touching $y$ .

Answer to Problem 47AYU

  

Explanation of Solution

Given information:

If $\tan \theta =a\tan \frac{\theta }{3}$ , express $\tan \frac{\theta }{3}$ in terms of $a$ .

On a touch −tone phone, each button produces a unique sound. The sound produced is the sum of two tones, given by

  $y=\sin \left(2\pi /t\right)$ and $y=\sin \left(2\pi ht\right)$

Where $l$ and $h$ are the low and high frequencies (cycles per second) shown in the illustration. For example, if you touch $7$ , the low frequency is $l=852$ cycles per second and the high frequency is $h=1209$ cycles per second. The sound emitted by touching $7$ is

  $y=\sin \left[2\pi \left(852\right)t\right]+\sin \left[2\pi \left(1209\right)t\right]$

  

Graph the sound emitted by touching $7$ .

Calculation:

The Graph of sound emitted by touching $7$ .

  

  $y=2\sin \left(2061\pi t\right).\cos \left(357\pi t\right)$

Hence, $y=2\sin \left(2061\pi t\right).\cos \left(357\pi t\right)$ .