solve the equation on the interval ( 0 leq \theta<2 pi ).\n( 4 cos ^{2} \theta=3 )\nwhat are the solutions…

solve the equation on the interval ( 0 leq \theta<2 pi ).\n( 4 cos ^{2} \theta=3 )\nwhat are the solutions in the interval ( 0 leq \theta<2 pi ) ? select the correct choice and fill in any answer boxes in your choice below.\na. the solution set is {}\n(simplify your answer. type an exact answer, using ( pi ) as needed. type your answer in radians. use integers or fractions for any numbers in the expression as needed.)\nb. there is no solution.

solve the equation on the interval ( 0 leq \theta<2 pi ).\n( 4 cos ^{2} \theta=3 )\nwhat are the solutions in the interval ( 0 leq \theta<2 pi ) ? select the correct choice and fill in any answer boxes in your choice below.\na. the solution set is {}\n(simplify your answer. type an exact answer, using ( pi ) as needed. type your answer in radians. use integers or fractions for any numbers in the expression as needed.)\nb. there is no solution.

Answer

Explanation:

Step1: Solve for (\cos\theta)

Given (4\cos^{2}\theta = 3), divide both sides by (4): (\cos^{2}\theta=\frac{3}{4}). Take the square - root of both sides: (\cos\theta=\pm\frac{\sqrt{3}}{2}).

Step2: Find (\theta) values

When (\cos\theta=\frac{\sqrt{3}}{2}), (\theta = \frac{\pi}{6}+2k\pi) or (\theta=\frac{11\pi}{6}+2k\pi) ((k\in\mathbb{Z})). In the interval (0\leq\theta < 2\pi), when (k = 0), (\theta=\frac{\pi}{6},\frac{11\pi}{6}). When (\cos\theta=-\frac{\sqrt{3}}{2}), (\theta=\frac{5\pi}{6}+2k\pi) or (\theta=\frac{7\pi}{6}+2k\pi) ((k\in\mathbb{Z})). In the interval (0\leq\theta < 2\pi), when (k = 0), (\theta=\frac{5\pi}{6},\frac{7\pi}{6}).

Answer:

(\frac{\pi}{6},\frac{5\pi}{6},\frac{7\pi}{6},\frac{11\pi}{6})