We are to show that the given parametric curve is a circle.
The trajectory of a circle with a radius r will satisfy the following relationship:
(with (x_c,y_c) being the center point)
We are given the x and y in a parametric form which can be further rewritten (using properties of sin/cos):
Squaring and adding both gives:
The last expression shows that the given parametric curve is a circle with the center (0,0) and radius A.
A beat is an interference pattern between two sounds of slightly different frequencies, perceived as a periodic variation in volume whose rate is the difference of the two frequencies. Frequency beat is equal to,
The reference frequency in our case would be 392Hz, and since there is the possibility of the upper and lower range for the amount of beats per second that the two possible frequencies are heard would be
Therefore the two possible frequencies the piano wire is vibrating at, would be 396Hz and 388Hz