Categorise the following function of time `: sin omega t + sin 3 omega t + sin 5 omega t ` as
(a) S.H.M.
(b) Periodic but no S.H.M. .Also give the period.
Hint `:` Find the smallest interval of time after which the sum of 3 terms repeats

Explanation

`sin omega t + cos omegat `
`= sin omegat + sin ((pi)/(2) - omegat )`
`= 2 sin ((pi)/(4)) cos (omegat-(pi)/(4))` ( using `sin C + sin D = 2 sin . (C+D)/(2) cos . (C-D)/(2))`
`= 2 xx (1)/( sqrt(2)) cos ( omegat - (pi)/( 4))`
`= sqrt(2) cos ( omegat -(pi)/( 4))`
This function represent a SHM having period `T= (2pi)/( omegat) ` and phase `( - (pi)/( 4)) `or `( (7pi)/(4))`