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Is cos8t-sin12t a periodic signal?

Yes

A signal whose instantaneous amplitude changes repeatedly with time is called a periodic signal. The general expression is: where t represents time and T represents period

Common periodic signals include: sinusoidal signals, pulse signals and their rectification, differentiation, integration, etc. This type can be called a simple signal. Their characteristics are that there are no more than two extreme points in a cycle and the periodic characteristics are obvious. For this type of signal with clear periodic characteristics, the judgment of whether it is periodic or not is relatively simple, and the period measurement methods are also very mature and perfect, such as zero-crossing detection method, pulse shaping method, etc.