Answer
The graph is shown below:
Work Step by Step
We know that the number 4 has been pressed repeatedly. The low frequency for 4 is $770$ cycles per second and the high frequency is $\text{1209}$ cycles per second. The sound produced by touching each button in the touch phone is described with the help of the formula given below:
$y=\sin 2\pi lt+\sin 2\pi ht$
The low frequency (l) is $770$ cycles per second and the high frequency (h) is $1209$ cycles per second. The sound produced by touching 4 is described as
$\begin{align}
& y=\sin 2\pi lt+\sin 2\pi ht \\
& =\sin 2\pi \left( 770 \right)t+\sin 2\pi \left( 1209 \right)t \\
& =\sin 1540\pi t+\sin 2418\pi t
\end{align}$
Now, to describe the repeated sound as a product of sines and cosines, the identity $\sin \alpha +\sin \beta =2\sin \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}$ is applied as shown below:
$\begin{align}
& y=\sin 1540\pi t+\sin 2418\pi t \\
& =2\sin \left( \frac{1540\pi t+2418\pi t}{2} \right)\cdot \cos \left( \frac{1540\pi t-2418\pi t}{2} \right) \\
& =2\sin 1,979\pi t\cdot \cos \left( -439 \right)\pi t \\
& =2\sin 1,979\pi t\cdot \cos 439\pi t
\end{align}$
The graph is shown below:
