Question

solve the equation in degrees for all exact solutions where appropriate. round approximate answers in degrees to the nearest tenth. write answers using the least possible nonnegative angle measures.
\\( \cos \frac{\theta}{2}=-\frac{1}{2} \\)

choose the correct answer below
\\( \bigcirc \\) a. the solution set is \\( \left\\{240^{\circ}+360^{\circ} n, 480^{\circ}+360^{\circ} n \\), where \\( n \\) is any integer\\}
\\( \bigcirc \\) b. the solution set is \\( \left\\{60.0^{\circ}+180^{\circ} n, 120.0^{\circ}+180^{\circ} n \\), where \\( n \\) is any integer\\}
\\( \bigcirc \\) c. the solution set is \\( \left\\{120^{\circ}+720^{\circ} n, 240^{\circ}+720^{\circ} n \\), where \\( n \\) is any integer\\}
\\( \bigcirc \\) d. the solution set is \\( \left\\{240^{\circ}+720^{\circ} n, 480^{\circ}+720^{\circ} n \\), where \\( n \\) is any integer\\}\\)

Explanation:

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

We know that \(\cos x = -\frac{1}{2}\) has solutions \(x = 120^{\circ}+ 360^{\circ}n\) or \(x = 240^{\circ}+ 360^{\circ}n\) for any integer \(n\) (since cosine is negative in the second and third quadrants and \(\cos120^{\circ}=-\frac{1}{2},\cos240^{\circ}=-\frac{1}{2}\)). Here \(x=\frac{\theta}{2}\), so \(\frac{\theta}{2}=120^{\circ}+ 360^{\circ}n\) or \(\frac{\theta}{2}=240^{\circ}+ 360^{\circ}n\).

Step2: Solve for \(\theta\)

Multiply both sides of each equation by 2. For \(\frac{\theta}{2}=120^{\circ}+ 360^{\circ}n\), we get \(\theta = 2\times(120^{\circ}+ 360^{\circ}n)=240^{\circ}+ 720^{\circ}n\). For \(\frac{\theta}{2}=240^{\circ}+ 360^{\circ}n\), we get \(\theta = 2\times(240^{\circ}+ 360^{\circ}n)=480^{\circ}+ 720^{\circ}n\).

Answer:

D. The solution set is \(\{240^{\circ}+ 720^{\circ}n, 480^{\circ}+ 720^{\circ}n\}\), where \(n\) is any integer\}