Determining total amplitude of out of phase waves

[Gza]

I understand there is a way to solve the problem with complex numbers, but am at a loss as to how:

Determine the amplitude of the resultant motion when two sinusoidal motions having the same frequency and traveling in the same direction are combined, if their amplitudes are 3 cm and 4 cm and they differ in phase by pi/2 radians.

 

[Tide]

I understand there is a way to solve the problem with complex numbers, but am at a loss as to how:

Determine the amplitude of the resultant motion when two sinusoidal motions having the same frequency and traveling in the same direction are combined, if their amplitudes are 3 cm and 4 cm and they differ in phase by pi/2 radians.

The amplitude will be 5!

Basically

A = 3 \sin X + 4 \cos X = 5 \left ( \frac {3}{5} \sin X + \frac {4}{5} \cos X \right)

where X is the time/space phase. Now just define \phi = \tan^{-1} \frac {4}{3} so that

A = 5 \sin \left( X+\phi \right)

 

[Gza]

You are a busy man tide! Thanks for the help.