Need explanation about sin rule

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Discussion Overview

The discussion revolves around the sine rule and its application in evaluating integrals involving sine functions. Participants are examining a specific term in an integral and questioning its contribution to the overall result.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant questions why a specific term is considered equal to zero in the context of an integral.
  • Another participant asserts that the term sin(wt)/6 x sin(2wt-(pi/4)) is not equal to zero.
  • A different participant suggests that while the term itself is not zero, its integral over a complete cycle contributes nothing, indicating a potential simplification in the evaluation process.
  • One participant proposes using the identity ##\sin(a)\sin(b) = \frac 1 2 \bigl(\cos(a-b) - \cos(a+b)\bigr)## as a method to demonstrate that the term integrates to zero.

Areas of Agreement / Disagreement

Participants express differing views on the treatment of the term in question, with no consensus reached on whether it should be dismissed or evaluated in full.

Contextual Notes

There are unresolved assumptions regarding the integral's limits and the specific conditions under which the term may be considered negligible.

MissP.25_5
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Hello.
This is part of a solution but I don't understand the underlined part. Why is it equals to 0?
 

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sin(wt)/6 x sin(2wt-(pi/4))
is not equal to zero.
 
MissP.25_5 said:
Hello.
This is part of a solution but I don't understand the underlined part. Why is it equals to 0?
That term itself isn't zero. But the area under it, taken over the complete cycle 0→2 Pi contributes nothing to the integral that you are about to evaluate. So the author is just looking ahead and seeing that he can save himself a bit of work here, and not bothering to process a term that is going to end up as zero anyway.

Why don't you work through without dismissing that term, and see how the result pans out?
 
An easy way to show that that term integrates to zero is to use the identity ##\sin(a)\sin(b) = \frac 1 2 \bigl(\cos(a-b) - \cos(a+b)\bigr)##.
 

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