Question

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

Answer 1

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

Explanation:

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)`.