How can I simplify adding two sine waves using trigonometric identities?

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SUMMARY

The discussion focuses on simplifying the addition of two sine waves represented by the equation y = Asin(wt) + Bsin(wt + x), where w = 2(pi)f and x is a phase shift. The method involves using the trigonometric identity sin(u + v) = sin(u)cos(v) + cos(u)sin(v) to rewrite sin(wt + x) in terms of sin(wt) and cos(wt). By collecting terms, the equation can be expressed as Csin(wt) + Dcos(wt), which can further be transformed into the form y = Rsin(wt + q) using the relationships C = Rcos(q) and D = Rsin(q), where R = sqrt(C^2 + D^2) and q = arctan(D/C).

PREREQUISITES
  • Understanding of trigonometric identities, specifically sin(u + v).
  • Familiarity with sine and cosine functions.
  • Basic knowledge of phase shifts in wave functions.
  • Ability to manipulate algebraic expressions involving trigonometric functions.
NEXT STEPS
  • Study the derivation and applications of the sine addition formula.
  • Explore the concept of phasors and their use in simplifying wave equations.
  • Learn about the implications of amplitude and phase in wave interference.
  • Investigate the use of R = sqrt(C^2 + D^2) in signal processing.
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Students and professionals in physics, engineering, and mathematics who are working with wave functions and trigonometric identities, particularly those involved in signal processing and wave analysis.

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Add the following sine waves

y= Asin(wt) + Bsin(wt + x) Where w=2(pi)f and x is a phase shift


I SIMPLIFIED this for my friend using rotating vectors, then i got really annoyed realizing that I didn't know how to do it the direct way (trig identities).

So I've posted it here for all you smart folks to try
 
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What do you mean by solved?
 
Ok, after consideration, is this what you want?

We may use sin(u+v)=sinucosv+cosusinv to rewrite sin(wt+x) in terms of cos(wt) and sin(wt), and then collecting terms we can rewrite the equation as Csinwt+Dcoswt for some choice of C and D. You may then blindly apply a formula whose derivation comes from :

suppose Csinz+Dcosz= Rsin(z+q), then C = Rcos(q) and D=Rsin(q) after applying that identity, from which we see R=sqrt(C^2+d^2) and tan(q) = D/C ie q=arctan(D/C)

So, you may conclude that y= Rsin(wt+q) for suitable R and q which can be derived with a little work. You can use that to do most anything you wish now.
 

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