Adding Sine and Cosine Waves- How to get formula

[opus]

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.

 

[opus]

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

 

[tnich]

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!

 

[opus]

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]

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.

 

[opus]

Thank you tnich.