Adding Sine and Cosine Waves- How to get formula

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  • #1
opus
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I have included a screenshot of a part of my textbook that is giving me a slight bit of confusion.

It's talking about how to get the formula for adding sines and cosines.

The part that I am confused about is the very first formula introduced in the screenshot.

From what I understand, we are taking one side of the sum of sines formula, and Asin(x)+Bcos(x).
The part in parentheses I do understand. It's stating the sum of sines formula in a different way. But I do not understand why the ##\sqrt{A^2+B^2}##, which would be the hypotenuse, is on the outside of the parentheses.
 

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  • #2
opus
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Here’s a shot of my understanding of it thus far.
 

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  • #3
tnich
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I have included a screenshot of a part of my textbook that is giving me a slight bit of confusion.

It's talking about how to get the formula for adding sines and cosines.

The part that I am confused about is the very first formula introduced in the screenshot.

From what I understand, we are taking one side of the sum of sines formula, and Asin(x)+Bcos(x).
The part in parentheses I do understand. It's stating the sum of sines formula in a different way. But I do not understand why the ##\sqrt{A^2+B^2}##, which would be the hypotenuse, is on the outside of the parentheses.
In a thread earlier today, you learned that you could multiply one side of an equation by 1 and still maintain equality. So multiply ##A~sin(x) + B~cos(x)## by ##\frac {\sqrt{A^2+B^2}} {\sqrt{A^2+B^2}}## and see what you get. It should look a lot like the first line of the derivation.
 
  • #4
opus
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AH! That's the one! Threw me off a little because it looks like they did that, and they factored the square root term out of the numerator and left it in the denominator. That is perfect, thank you!
 
  • #5
opus
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As an additional question, now that we have the Sum of Sines and Cosines formula
##Asin\left(x\right)+Bcos\left(x\right)## = ##\sqrt{A^2+B^2}sin\left(x+θ\right)##,
is the +θ considered a phase shift? That is, are we taking the graph of sin(x) and shifting it θ to the left?
 
  • #6
tnich
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As an additional question, now that we have the Sum of Sines and Cosines formula
##Asin\left(x\right)+Bcos\left(x\right)## = ##\sqrt{A^2+B^2}sin\left(x+θ\right)##,
is the +θ considered a phase shift? That is, are we taking the graph of sin(x) and shifting it θ to the left?
Yes, that is exactly the case.
 
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  • #7
opus
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Thank you tnich.
 

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