What is constant in $S.H.M.$?

What is the importance of periodic $\text{sine}$ and $\text{cosine}$ functions in physics?

The displacement of a particle varies according to the relation $x = 3 \sin 100t + 8 \cos^2 50t$. Which of the following is incorrect about this motion?

Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?
$(a)$ The rotation of Earth about its axis.
$(b)$ Motion of an oscillating mercury column in a $U$-tube.
$(c)$ Motion of a ball bearing inside a smooth curved bowl,when released from a point slightly above the lowermost point.
$(d)$ General vibrations of a polyatomic molecule about its equilibrium position.

The $S.H.M.$ of a particle is given by the equation $x = 2 \sin \omega t + 4 \cos \omega t$. Its amplitude of oscillation is ........ units.

Write the equation representing the relation between time period and frequency.