Question

How do you find the amplitude, period, phase shift given `y=-5sin3x+12cos3x`?

Answer 1

Amplitude = 13, period = `(2pi)/3` and phase shift `= - 0.6552`, nearly.

Explanation:

`y = -5 sin 3x + 12 cos 3x`

`= 13 ( (-5/13 )sin 3x + (12/13) cos 3x`

`13 ( sin 3x cos alpha + cos 3x sin alpha )`,

`alpha = pi - sin^(-1)(12/13)`

`= 13 sin ( 3x + alpha )`. Now, easily

amplitude = 13, period = `(2pi)/3` and

phase shift`= - alpha/3 = 1/3 ( sin^(12/13) - pi )`

`=1/3 ( 1. 1760 - 3.1416 ) = - 0.6552`, nearly.

See ( not-to-uniform-scale ) graph, with all these aspects:
graph{(y-13 sin ( 3x -1.9656))(y -13+0x )(y+13+0x)( x - 1/3pi+0.0001y)( x + 1/3pi+0.0001y)=0[-2 2 -16 16]}