Chapter 16.1, Problem 62E

Music Musical sounds exhibit the same kind of periodic behavior as the trigonometric functions. High-pitched notes have short periods (less than 1/1000 second) while the lowest audible notes have periods of about 1/100 second. Some electronic synthesizers work by superimposing (adding) sinusoidal functions of different frequencies to create different textures. Exercises 59–62 show some examples of how superposition can be used to create interesting periodic functions.

Discord If we add two sinusoidal functions with similar, but unequal, frequencies, the result is a function that “pulsates,” or exhibits “beats.” (Piano tuners and guitar players use this phenomenon to help them tune an instrument.) Graph the function

$y=\cos x+\cos \left(0.9x\right)$ in the window $-50\le x\le 50$

and $-2\le y\le 2$, and estimate the period of the resulting wave.