Need help understanding how an example from my textbook represented a trigonometric fourier series as a compact trigonometric fourier series..... I will type the example out in the comments

Question

chrisplusian · Answer

A trigonometric fourier series is represented:
x(t)=2+3cos(2t)+4sin(2t)+2sin(3t+30)-cos(7t-150)

The Compact form is :
3cos(2t)+4sin(2t)=5cos(2t-53.13)

*AS a side note the phase shifts here are in degrees

chrisplusian · Answer

I just want to understand how they manipulated one into the other

thomas5267 · Answer

$$
A\cos(a+b)=A\cos(a)\cos(b)-A\sin(a)\sin(b)\
A\cos(2t+b)=A\cos(2t)\cos(b)-A\sin(2t)\sin(b)\
A\cos(b)=3\
A\sin(b)=4\
\tan(b)=\frac{4}{3}\
A^2\cos^2(b)+A^2\sin^2(b)=3^2+4^2=25\
A=5
$$