Why does f[x]=Sin[2x]+Sin[3x] Not Equal a Trig Function?

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SUMMARY

The function f[x] = Sin[2x] + Sin[3x] does not represent a single trigonometric function due to the differing frequencies of its components. Specifically, when combining two harmonic waves with distinct frequencies, such as A1*Sin(ω1*t+φ1) and B1*Sin(ω2*t+φ2) where ω1 ≠ ω2, the resultant function cannot be expressed in the standard form of a simple harmonic wave ASin(ωt+φ). This conclusion is supported by the definition of harmonic functions and their properties in wave theory.

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  • Understanding of harmonic functions and their definitions
  • Knowledge of trigonometric identities and properties
  • Familiarity with wave mechanics and frequency concepts
  • Basic calculus for analyzing wave functions
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  • Explore the concept of superposition in wave theory
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  • Investigate the implications of frequency differences in wave interference
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Students of physics, mathematicians, and anyone interested in wave mechanics and trigonometric function analysis will benefit from this discussion.

skyriver
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for example
if f[x]=Sin[2x]+Sin[3x] then f[x] can not be a Trigonometric function,
why? how many ways to prove this?
 
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You used the word "harmonic" in your title and "trigonometric" in the text. Would you give the definition of those words that you are using?
 


What i mean by a simple harmonic wave is a function which has this form f(t)=ASin[ωt+φ], where A is the amplitude, ω is the frequency, and φ is the phase,t is time.

my question is :if two two harmonic waves A1*Sin(ω1*t+φ1) and B1*Sin(ω2*t+φ2) have different frequencys ,i.e ω1 is not equal to ω2, then the sum of these two functions can not be any harmonic wave ,i.e the sum can not be function like ASin(ωt+φ) .
 

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