Part A.
The length of the non-compressed spring can be determined by setting

Now, evaluating the given function at θ=0, we get:
![f(0)=2\cos (0)+\sqrt[]{3}.](https://img.qammunity.org/2023/formulas/mathematics/college/rnbhuqg5oy8xwrzizsk4qay8yg06sbj4y3.png)
Recall that:

Therefore:
![f(0)=2\cdot1+\sqrt[]{3}=2+\sqrt[]{3.}](https://img.qammunity.org/2023/formulas/mathematics/college/ff5ini27w0l25zk39bk7bhwj71r98ea4u6.png)
Now, to determine all the possible times at which the length will be equal to the non-compressed length, we set the following equation:
![2\cos \theta+\sqrt[]{3}=2+\sqrt[]{3}.](https://img.qammunity.org/2023/formulas/mathematics/college/eax5679zxbog3drefyo75c5ii145vm4ms1.png)
Solving the above equation for θ, we get:

The solutions to the above equation are:

Answer part A:

Part B: If we set θ to the double of itself, then, the solutions to the following equation
![f(2\theta)=2+\sqrt[]{3},](https://img.qammunity.org/2023/formulas/mathematics/college/46lk8xbqd4oiycxulm19dr19amxbcvbi01.png)
are:

therefore:

If

then

Answer part B: The difference is that the period is half of the original period.
