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 non - negative angle measures.
cos \frac { \theta } { 2 } = \frac { sqrt { 3 } } { 2 }
choose the correct answer below.
a. the solution set is ( left{ 15.0 ^ { circ } + 180 ^ { circ } n, 165.0 ^ { circ } + 180 ^ { circ } n ), where ( n ) is any integer).
b. the solution set is ( left{ 60 ^ { circ } + 720 ^ { circ } n, 660 ^ { circ } + 720 ^ { circ } n ), where ( n ) is any integer).
c. the solution set is ( left{ 30 ^ { circ } + 720 ^ { circ } n, 330 ^ { circ } + 720 ^ { circ } n ), where ( n ) is any integer).
d. the solution set is ( left{ 60 ^ { circ } + 360 ^ { circ } n, 660 ^ { circ } + 360 ^ { circ } n ), where ( n ) is any integer).

Explanation:

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

We know that \(\cos x=\frac{\sqrt{3}}{2}\) has solutions \(x = 30^{\circ}+360^{\circ}n\) or \(x=330^{\circ}+360^{\circ}n\), where \(n\) is any integer.
Since \(x = \frac{\theta}{2}\), we have \(\frac{\theta}{2}=30^{\circ}+360^{\circ}n\) or \(\frac{\theta}{2}=330^{\circ}+360^{\circ}n\).

Step2: Solve for \(\theta\)

Multiply both sides of \(\frac{\theta}{2}=30^{\circ}+360^{\circ}n\) by \(2\) to get \(\theta = 60^{\circ}+720^{\circ}n\).
Multiply both sides of \(\frac{\theta}{2}=330^{\circ}+360^{\circ}n\) by \(2\) to get \(\theta=660^{\circ}+720^{\circ}n\).

Answer:

B. The solution set is \(\{60^{\circ}+720^{\circ}n,660^{\circ}+720^{\circ}n\), where \(n\) is any integer\(\}\)