# Chapter 5 - Section 5.4 - Product-to-Sum and Sum-to-Product Formulas - Exercise Set - Page 691: 62 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: 