How do you find the amplitude, period, phase shift given #y=3sinpix-5cospix#?

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Answer 1 · 2016-10-17T10:09:53

Amplitude is #sqrt34#, period is #2# and phase shift is #1/pitan^(-1)(-5/3)#.

We have #y=3sinpix-5cospix# (Note #3^2+5^2=34#)

= #sqrt34(3/sqrt34sinpix-5/sqrt34cospix)#

= #sqrt34(sinpixcosalpha-cospixsinalpha)#

= #sqrt34sin(pix-alpha)#,

where, as #sinalpha=5/sqrt34# and #cosalpha=3/sqrt34#,

#alpha=tan^(-1)(-5/3)#

Now, as in #y=rsin(px+q)#

amplitude is #r#, period is #(2pi)/p# and phase shift is #-q/p#.

In #y=3sinpix-5cospix=sqrt34sin(pix-alpha)#

amplitude is #sqrt34#, period is #(2pi)/pi=2# and phase shift is #alpha/pi=1/pitan^(-1)(-5/3)#.