Question

Use a trigonometry formula to write the two terms as a single harmonic. Find the period and amplitude. Compare computer plots of your result and the given problem.

$\mathit{c}\mathit{o}\mathit{s}\mathit{\pi }\mathit{x}\mathbf{-}\mathit{c}\mathit{o}\mathit{s}\mathit{\pi }\left(x-1/2\right)$

Short answer

The period and amplitude are T = 2 and $A=\sqrt{2}$.

Step-by-step solution

Given information

A fundamental musical tone and some of its overtones of harmonics are:

$cos\pi x-cos\pi (\mathrm{x}-1/2)$

Formula of trigonometry

Trigonometry formula is$\mathbf{[}\mathbf{sin}\mathbf{}\mathbf{a}\mathbf{+}\mathbf{sin}\mathbf{}\mathbf{b}\mathbf{}\mathbf{=}\mathbf{2}\mathbf{sin}\frac{\mathbf{a}\mathbf{+}\mathbf{b}}{\mathbf{2}}\mathbf{cos}\frac{\mathbf{a}\mathbf{-}\mathbf{b}}{\mathbf{2}}\mathbf{]}$.

Write the two terms as a single harmonic

The given harmonic term is$cos\pi x-cos\pi (\mathrm{x}-1/2)$.

$$\begin{gathered} =-2\sin \frac{\pi x+\pi x-\pi /2}{2}\sin \frac{\pi x+\pi x-\pi /2}{2} \\ =-2\sin \left(\pi x-\frac{\pi }{4}\right)\sin \frac{\pi }{4} \\ =-\sqrt{2}\sin \pi \left(x-\frac{1}{4}\right) \end{gathered}$$

Thus,the period and amplitude are T = 2 and $A=\sqrt{2}$.

The computer plots of the start and end functions

The plot of start of the function is shown below

And, plot of function is shown below.end of the

Thus, the period and amplitude are T = 2 and $A=\sqrt{2}$.