The sum of two sinusoids is periodic provided that their frequencies are______ of a fundamental frequency.

The sum of two sinusoids is periodic provided that their frequencies are multiples (or integer ratios) of a fundamental frequency.

Here's a step-by-step explanation:

1. A sinusoid is a mathematical curve that describes a smooth periodic oscillation. It is defined by the function y = A sin(B(x - C)) + D, where A is the amplitude, B is the frequency, C is the phase shift, and D is the vertical shift.

2. The frequency of a sinusoid is the number of cycles it completes in a given amount of time. It is usually measured in hertz (Hz), which is equivalent to cycles per second.

3. If you have two sinusoids, they will be periodic (i.e., they will repeat their pattern over time) if their frequencies are multiples of a fundamental frequency. This means that the frequency of one sinusoid is a whole number times the frequency of the other.

4. For example, if the frequency of the first sinusoid is 2 Hz and the frequency of the second sinusoid is 3 Hz, the sum of the two sinusoids will be periodic because 2 and 3 are both multiples of 1, which can be considered the fundamental frequency in this case.

5. In general, the sum of two sinusoids will be periodic if their frequencies are in the ratio of two integers. This is because the sum of two sinusoids with frequencies that are integer multiples of a fundamental frequency will complete a whole number of cycles over the same period of time, resulting in a repeating pattern.