solve for exact solutions over the interval 0°,360°)\n\n\\( \\sin \\frac { \\theta } { 2 } = 0 \\)\n\nselect…

solve for exact solutions over the interval 0°,360°)\n\n\\( \\sin \\frac { \\theta } { 2 } = 0 \\)\n\nselect the correct choice below and, if necessary, fill in the\nanswer box to complete your choice.\n\na. the solution set is \\( \\{ \\square ^ { \\circ } \\} \\)\n(simplify your answer. use a comma to separate\nanswers as needed. do not include the degree symbol\nin your answer.)\nb. the solution set is \\( \\varnothing \\)

solve for exact solutions over the interval 0°,360°)\n\n\\( \\sin \\frac { \\theta } { 2 } = 0 \\)\n\nselect the correct choice below and, if necessary, fill in the\nanswer box to complete your choice.\n\na. the solution set is \\( \\{ \\square ^ { \\circ } \\} \\)\n(simplify your answer. use a comma to separate\nanswers as needed. do not include the degree symbol\nin your answer.)\nb. the solution set is \\( \\varnothing \\)

Answer

Explanation:

Step1: Solve for (\frac{\theta}{2})

We know that if (\sin x = 0), then (x = k\cdot180^{\circ}), where (k\in\mathbb{Z}). For (\sin\frac{\theta}{2}=0), we have (\frac{\theta}{2}=k\cdot180^{\circ}).

Step2: Solve for (\theta)

Multiply both sides of (\frac{\theta}{2}=k\cdot180^{\circ}) by (2) to get (\theta = k\cdot360^{\circ}).

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

When (k = 0), (\theta=0); when (k = 1), (\theta = 360^{\circ}) (but (360^{\circ}) is not in the interval ([0^{\circ},360^{\circ}))).

Answer:

A. The solution set is (0)