Understanding the Amplitude of sin2x

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SUMMARY

The amplitude of the function sin2x is determined by the coefficient in front of sinx when expressed in the form 2sinxcosx. The time period of sin2x is π, while its amplitude varies with the function cosx, which is not a constant. In simple harmonic motion (SHM) and wave motion, the amplitude is defined as the maximum value of the wave function, which in this case is influenced by the varying cosx term. Therefore, the amplitude of sin2x is not constant but changes based on the value of cosx.

PREREQUISITES
  • Understanding of trigonometric functions, specifically sine and cosine.
  • Familiarity with the concept of amplitude in wave motion.
  • Knowledge of the time period of periodic functions.
  • Basic principles of simple harmonic motion (SHM).
NEXT STEPS
  • Study the properties of trigonometric identities, particularly the product-to-sum formulas.
  • Explore the concept of amplitude in more complex waveforms, including beats and interference.
  • Learn about the graphical representation of trigonometric functions and their transformations.
  • Investigate the relationship between wave numbers and amplitude in wave mechanics.
USEFUL FOR

Students of mathematics and physics, particularly those studying wave motion and trigonometric functions, as well as educators seeking to clarify concepts related to amplitude and periodicity.

Apurv Zoad
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If we look at the graph of sin2x, well its Time Period is pi, and amplitude is same as that of sinx. But if we expand sin2x as 2sinxcosx, rewriting it as 2cosxsinx. According to the Physics in SHM, Wave Motion, whatever is infront of sinx is called as Amplitude. Now, y = 2cosxsinx = Asinx, then why did Amplitude rise by A times, that is by 2cosx times ? I think that the cosx is the decreasing function, that's why the Amplitude must remain same !
 
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The amplitude ##A## must be a constant. ##\cos x## changes with ##x## so isn't an amplitude.
 
Only if one has two trigonometric functions, say ##\sin(k_1 x+b_1)## and ##\sin(k_2 x + b_2)##, that have wavenumbers of very different magnitudes, ##k_2 >> k_1## we can call a factor of ##\sin(k_1 x+b_1)## in front of ##\sin(k_2 x + b_2)## an "amplitude" because of the phenomenon of beats. If ##k_1## and ##k_2## are of same order of magnitude, this is not appropriate.
 
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