How to Find the Period of x(t) = sin(5t) - 4cos(7t) in Signals and Systems?

The discussion addresses how to find the period of a composite continuous-time signal x(t) = sin(5t) - 4cos(7t) in signals and systems. The individual periods of the components sin(5t) and cos(7t) are derived from their angular frequencies (5 and 7 rad/s), yielding T1 = 2π/5 and T2 = 2π/7 respectively. To find the overall period of x(t), one must determine the least common multiple (LCM) of these individual periods. This involves finding the smallest positive time t where both sin(5t) and cos(7t) complete an integer number of cycles simultaneously. The LCM of the denominators 5 and 7 is 35, leading to the fundamental period T = 2π/35. The approach can also be understood by converting angular frequencies to frequencies (f = ω/2π), calculating individual periods as reciprocals of frequencies, and then finding the LCM of these periods after appropriate scaling to integers. This method ensures the composite signal's periodicity is correctly identified.
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