solve the equation for exact solutions over the interval 0°,360°). 2cos2θ = -√3 select the correct choice…
solve the equation for exact solutions over the interval 0°,360°). 2cos2θ = -√3 select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the solution set is { } (type an integer or a decimal. type your answer in degrees. do not include the degree symbol in your answer. use a comma to separate answers as needed.) b. the solution is the empty set
Answer
Explanation:
Step1: Isolate (\cos2\theta)
Divide both sides of the equation (2\cos2\theta = -\sqrt{3}) by (2). (\cos2\theta=-\frac{\sqrt{3}}{2})
Step2: Find the general solutions for (2\theta)
We know that (\cos\alpha =-\frac{\sqrt{3}}{2}) when (\alpha = 150^{\circ}+ 360^{\circ}n) or (\alpha=210^{\circ}+360^{\circ}n), where (n\in\mathbb{Z}). So, (2\theta = 150^{\circ}+360^{\circ}n) or (2\theta=210^{\circ}+360^{\circ}n)
Step3: Solve for (\theta)
Divide each equation by (2). For (2\theta = 150^{\circ}+360^{\circ}n), (\theta = 75^{\circ}+180^{\circ}n) For (2\theta=210^{\circ}+360^{\circ}n), (\theta = 105^{\circ}+180^{\circ}n)
Step4: Find solutions in the interval ([0,360))
When (n = 0): (\theta=75^{\circ}) (from (\theta = 75^{\circ}+180^{\circ}\times0)) and (\theta = 105^{\circ}) (from (\theta=105^{\circ}+180^{\circ}\times0)) When (n = 1): (\theta=75^{\circ}+180^{\circ}=255^{\circ}) and (\theta=105^{\circ}+180^{\circ}=285^{\circ}) When (n = 2), (\theta=75^{\circ}+360^{\circ}=435^{\circ}) (exceeds (360^{\circ})) and (\theta=105^{\circ}+360^{\circ}=465^{\circ}) (exceeds (360^{\circ}))
Answer:
A. The solution set is (75,105,255,285)