How Do You Calculate the Combined Phase Angle Shift of Multiple Waves?

[armolinasf]

Homework Statement

If I have three different waves each with difference amplitude and phase angle how would i find the combined phase angle shift?

wave 1: A=5 phi=0
wave 2: A=5 phi=pi/4
wave 3: A=9 phi=pi/2

Homework Equations

phi'=arctan(a2sin(phi)/(a1+a2cos(phi)), where phi in this case is the difference between the phase angles of two waves.

The Attempt at a Solution

My first strategy was to add wave 1 and 2 and then add their sum to wave three. this gave me the correct combined amplitude of 15.29. I then applied the same thinking to the phase angles but the answer i got was incorrect so i tried vector addition and using the law of cosines and I got a nonsensical answer - arcos(-1.41). Needless to say i need a nudge in the right direction. Thanks for the help

 

[Redbelly98]

Think of the (A,Φ) values as polar coordinates of vectors. Convert them to (x,y) coordinates, and then it's standard vector addition from there.