Example $1.$ The phase angle $$ describes the relative shift in the sinusoidal vibration of the spring$-$mass system resulting from the initial displacement, $x_0.$ Verify that equation

$x(t)=\sqrt[\phantom{2}]{x_0^2+\frac{v_0^2}{{}_n^2}}sin({}_{n}t+ta{n}^{-1}(\frac{x_0{}_n}{v_0}))$

condition $x(0)=x_0.$