Question

What is the amplitude and period of `x=asin(nt)+bcos(nt)`?

Answer 1

Amplitude is `sqrt(a^2+b^2)` and period is `(2pi)/n`.

Explanation:

As `x=asin(nt)+bcos(nt)`, we can write ir as `x=bcos(nt)+asin(nt)`

Now let `cosalpha=a/sqrt(a^2+b^2)` and `sinalpha=b/sqrt(a^2+b^2)`

Observe that as `|a/sqrt(a^2+b^2)|<1` and `|b/sqrt(a^2+b^2)|<1`, it is possible as `cos^2alpha+sin^2alpha=1`

Then `x=sqrt(a^2+b^2)[cos(nt)cosalpha+sin(nt)sinalpha]`

= `sqrt(a^2+b^2)cos(nt-alpha)`

Hence amplitude is `sqrt(a^2+b^2)` and period is `(2pi)/n`.