Proof of Periodic Sinusoidal waveforms

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SUMMARY

The discussion focuses on proving that the sinusoidal waveform x(t) = cos(2t + π/4) is periodic. The key to establishing periodicity lies in demonstrating that the function repeats its values over a specific interval. The periodic nature of cosine functions is confirmed by the equation cos(A + 2πn) = cos(A), which indicates that the function will return to its original value after an interval of 2π. Thus, the waveform is periodic with a fundamental period of π.

PREREQUISITES
  • Understanding of sinusoidal functions and their properties
  • Familiarity with trigonometric identities, specifically cosine and sine transformations
  • Knowledge of periodicity in mathematical functions
  • Basic skills in manipulating algebraic expressions involving angles
NEXT STEPS
  • Study the properties of periodic functions in trigonometry
  • Learn about the derivation of the fundamental period of sinusoidal waveforms
  • Explore trigonometric identities and their applications in proving periodicity
  • Investigate the relationship between phase shifts and periodicity in sinusoidal functions
USEFUL FOR

Students studying trigonometry, mathematicians interested in waveforms, and anyone looking to understand the periodic nature of sinusoidal functions.

tomh
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Homework Statement


Hi,

Have completely forgotten how to prove that a sinusoidal waveform is periodic and can't seem to find it anywhere. So was hoping someone could here.

I've got the signal x(t)=cos(2t+pi/4)

and am trying to prove it is periodic.

Homework Equations



wt=theta

f(x+k)=f(x)

cos(x)=sin(x+pi/2)

The Attempt at a Solution



I know that it is periodic, but to prove I think I have to prove that x(t)=0

Have tried letting t=-pi/8 but with the phase change of pi/2 to convert it to a sine again it gives me a non zero answer.

Anyone with any help would be greatly appreciated.
 
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welcome to pf!

hi tomh! welcome to pf! :smile:

(have a pi: π and a theta: θ :wink:)

cos(A + 2πn) = cosAcos2πn - sinAsin2πn = cosA

does that help?​
 

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