A particle oscillates according to the law
`s=(1+cos^(2)t+cos^(4)t)sin500t`
Expand this motion into its harmonic components and plot its spectrum.

Explanation

We have
`1+cos^(2)t+cos^(4)t=1+(1)/(2)(1+cos2t)+(1)/(4)(1+cos2t)^(2)=`
`=(1)/(4)[7-4cos2t+(1)/(2)(1cos4t)]=(1)/(8)(15+8cos2t+cos4t)`
Hence
`s=(1)/(8)(15-8cos2t-cos4t)sin500t(15)/(8)sin500t+cos2tsin500t+`
`+(1)/(8)cos4t.sin500t=(15)/(8)sin500t+(1)/(2)sin502t+(1)/(2)sin498+(1)/(16)sin504t+(1)/(16)sin496t`
The spectrum of the vibrations considered above is shown in Fig