Part A)
Since the function f represents the difference in length of the spring from its non-compressed position, then, when the spring is not compressed, the value of f(θ) is equal to 0:

Replace the expression for f(θ) and solve for θ:
![\begin{gathered} 2\cos (\theta)+\sqrt[]{3}=0 \\ \Rightarrow2\cos (\theta)=-\sqrt[]{3} \\ \Rightarrow\cos (\theta)=-\frac{\sqrt[]{3}}{2} \\ \Rightarrow\theta=\arccos (-\frac{\sqrt[]{3}}{2}) \\ \\ \therefore\theta=\pm(5)/(6)\pi+2\pi k;k\in\Z \end{gathered}](https://img.qammunity.org/2023/formulas/mathematics/college/88h0jegbosiy2b36ongehoxehvpfwq985m.png)
Part B)
Replace θ=2θ and find the solutions in the given interval:

For k=0, the positive solution lies in the interval [0,2π):

For k=1, both solutions lie in the interval [0,2π):

For f=2, the negative solution lies in the interval [0,2π):

Then, the solutions in the interval [0,2π) are:
