91Ó°ÊÓ

amplitude $a=4.0cm$

frequency$f=4.0Hz$

The wave equation is in the form

$x\left(t\right)=a\cos \left(\mathrm{Ó\neg }t\right)$

The angular frequency is in the form

localid="1650045287866" $\mathrm{Ó\neg }=2\mathrm{Ï€´Ú}=2\mathrm{Ï€}\left(4\right)=8\mathrm{Ï€}rad/s$

The initial condition is at $t=0s$, the object passes through the equilibrium to the right.

$x(t=0s)=a\cos \left(\mathrm{θ}\right)=0$

In order the cosine term to be zero it should satisfy the following condition:

$\mathrm{θ}=\frac{n\mathrm{π}}{2}$

The object is moving right in positive direction, so its velocity is positive at $t=0s$. The velocity function is the derivative of the position function:

$$\begin{gathered} v\left(t\right)=-\mathrm{Ó\neg }a\sin (\mathrm{Ó\neg }t+\mathrm{θ}) \\ v(t=0s)=-\mathrm{Ó\neg }a\sin \left(\mathrm{θ}\right)>0 \end{gathered}$$

Then, $\sin \mathrm{θ}$should be negative, therefore$\mathrm{θ}$should be negative

The equation now takes the form of substituting the values of angular frequency

localid="1650045341161" $x\left(t\right)=(4.0cm)×\cos \left(8\mathrm{π}t-\frac{\mathrm{π}}{2}\right)$