solve for exact solutions over the interval 0°,360°).\n\ncos(\\theta/2)=-1/2\n\nselect the correct choice…

solve for exact solutions over the interval 0°,360°).\n\ncos(\\theta/2)=-1/2\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the solution set is {□°}\n(simplify your answer. use a comma to separate answers as needed. do not include the degree symbol in your answer.)\nb. the solution set is ∅

solve for exact solutions over the interval 0°,360°).\n\ncos(\\theta/2)=-1/2\n\nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\n\na. the solution set is {□°}\n(simplify your answer. use a comma to separate answers as needed. do not include the degree symbol in your answer.)\nb. the solution set is ∅

Answer

Explanation:

Step1: Find the general solutions for (\frac{\theta}{2})

We know that if (\cos x =-\frac{1}{2}), then (x = 120^{\circ}+ 360^{\circ}n) or (x = 240^{\circ}+360^{\circ}n), where (n\in\mathbb{Z}). Since (x = \frac{\theta}{2}), we have (\frac{\theta}{2}=120^{\circ}+360^{\circ}n) or (\frac{\theta}{2}=240^{\circ}+360^{\circ}n).

Step2: Solve for (\theta)

For (\frac{\theta}{2}=120^{\circ}+360^{\circ}n), multiply both sides by (2) to get (\theta = 240^{\circ}+720^{\circ}n). For (\frac{\theta}{2}=240^{\circ}+360^{\circ}n), multiply both sides by (2) to get (\theta=480^{\circ}+720^{\circ}n).

Step3: Find solutions in the interval ([0^{\circ},360^{\circ}))

When (n = 0) for (\theta = 240^{\circ}+720^{\circ}n), (\theta=240^{\circ}). When (n=- 1) for (\theta = 480^{\circ}+720^{\circ}n), (\theta=480^{\circ}-720^{\circ}=-240^{\circ}) (rejected as it is not in ([0^{\circ},360^{\circ}))). When (n = 0) for (\theta = 480^{\circ}+720^{\circ}n), (\theta = 480^{\circ}) (rejected as it is not in ([0^{\circ},360^{\circ}))).

Answer:

A. The solution set is ({240})