# Adding Sine and Cosine Waves- How to get formula

• B
Gold Member
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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Gold Member
Here’s a shot of my understanding of it thus far.

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tnich
Homework Helper
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.

opus
Gold Member
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!

Gold Member
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?

tnich
tnich
Homework Helper
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.

jedishrfu and opus
Gold Member
Thank you tnich.

jedishrfu