solve the equation on the interval 0,2π).\ncos (2θ)= -√3/2\nwhat are the solutions to cos (2θ)= -√3/2 in the…

solve the equation on the interval 0,2π).\ncos (2θ)= -√3/2\nwhat are the solutions to cos (2θ)= -√3/2 in the interval 0,2π)? select the correct choice and fill in any answer boxes in your choice below.\na. θ=□ (simplify your answer. type an exact answer, using π as needed. type your answer in radians. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)\nb. there is no solution.

solve the equation on the interval 0,2π).\ncos (2θ)= -√3/2\nwhat are the solutions to cos (2θ)= -√3/2 in the interval 0,2π)? select the correct choice and fill in any answer boxes in your choice below.\na. θ=□ (simplify your answer. type an exact answer, using π as needed. type your answer in radians. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)\nb. there is no solution.

Answer

Explanation:

Step1: Find the general solution for ( \cos(2\theta)=-\frac{\sqrt{3}}{2} )

We know that ( \cos x = -\frac{\sqrt{3}}{2} ) when ( x=\frac{5\pi}{6}+2k\pi ) or ( x = \frac{7\pi}{6}+2k\pi ), ( k\in\mathbb{Z} ). Since ( x = 2\theta ), then ( 2\theta=\frac{5\pi}{6}+2k\pi ) or ( 2\theta=\frac{7\pi}{6}+2k\pi ). Solving for ( \theta ), we get ( \theta=\frac{5\pi}{12}+k\pi ) or ( \theta=\frac{7\pi}{12}+k\pi ), ( k\in\mathbb{Z} ).

Step2: Find the solutions in the interval ( [0,2\pi) )

When ( k = 0 ): ( \theta=\frac{5\pi}{12} ) or ( \theta=\frac{7\pi}{12} ) When ( k = 1 ): ( \theta=\frac{5\pi}{12}+\pi=\frac{5\pi + 12\pi}{12}=\frac{17\pi}{12} ) or ( \theta=\frac{7\pi}{12}+\pi=\frac{7\pi+12\pi}{12}=\frac{19\pi}{12} ) When ( k = 2 ), ( \theta=\frac{5\pi}{12}+2\pi=\frac{5\pi + 24\pi}{12}=\frac{29\pi}{12}>2\pi ) and ( \theta=\frac{7\pi}{12}+2\pi=\frac{7\pi + 24\pi}{12}=\frac{31\pi}{12}>2\pi )

Answer:

( \theta=\frac{5\pi}{12},\frac{7\pi}{12},\frac{17\pi}{12},\frac{19\pi}{12} )