How can the Euler formula be used to derive the properties of rotating vectors?

[Fjolvar]

I copied a diagram from my book of rotating vectors, and I just want to know how they got the following:

A cos(theta-phi) = A(cos(theta)cos(phi)+sin(theta)sin(phi))

and

A sin(theta-phi) = A(sin(theta)cos(phi)-cos(theta)sin(phi))

Which properties were used?

Thanks.

 

[mathman]

These are basic equations, that hold for sines and cosines in general, which are taught in any trigonometry course.

Note: there is an error in your cosine expression. One of the - signs should be +. You have to change one sign, but not both.

 

[Fjolvar]

These are basic equations, that hold for sines and cosines in general, which are taught in any trigonometry course.

Note: there is an error in your cosine expression. One of the - signs should be +. You have to change one sign, but not both.

Thanks, I meant to write a + sign. So I guess I'll just understand this as an identity. I'm still curious how it's proven.

 

[mathman]

If you are familiar with the Euler formula [e^ix = cosx + isinx] you can derive it very quickly using x = a + b.