The algebra of angles

Adding angles

We might have a sine-wave with a phase shift and want to turn it into a sine wave plus a cosine wave.

So do you remember the formula to work out sin(a + b) ?

    sin(a + b) = sin(a) cos (b) + cos(a) sin(b)

We also have

    cos(a + b) = cos(a) cos(b) - sin(a) sin(b)

(The easy way to prove these is with complex exponentials!)


So a sine wave with phase shift b,

    A sin(at + b)
 
can be written instead as

    C sin(at) + D cos(at)

where

    C = A cos(b)

and

    D = A sin(b).


Now if instead you want to express a sum of sines and cosines as a phase shift, you can work out A from C and D by

    A² = C² + D²

since   cos²(b) + sin²(b) =1, and

b = tan^-1(D/C)

Notation

You may need to read these notes again, substituting omega for a and phi for b.

Take care!

Many textbooks solve problems by the phase shift method.  If you can grit your teeth and regard the sines and cosines as parts of a complex number, the arithmetic is VERY much easier!
Have a look at this.