# Adding Sine and Cosine Waves- How to get formula

• B
• opus
In summary, the conversation discusses a screenshot from a textbook that explains how to get the formula for adding sines and cosines. The person is confused about the placement of the hypotenuse in the formula and learns that it is a result of multiplying by 1 in order to maintain equality. The conversation also delves into whether the +θ in the formula is a phase shift, and it is confirmed that it is.

#### opus

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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Here’s a shot of my understanding of it thus far.

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opus said:
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
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!

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
opus said:
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
Thank you tnich.

jedishrfu