solve the equation on the interval ( 0 leq \theta < 2pi ).\n sec \frac{3\theta}{2} = 2 \nwhat are the…

solve the equation on the interval ( 0 leq \theta < 2pi ).\n sec \frac{3\theta}{2} = 2 \nwhat are the solutions to ( sec \frac{3\theta}{2} = 2 ) in the interval ( 0 leq \theta < 2pi )? select the correct choice and fill in any answer\n\na. the solution set is\n(simplify your answer. type an exact answer, using ( pi ) as needed. type your answer in radians. use answers as needed.)\nb. there is no solution.

Answer

Explanation:

Step1: Use the reciprocal identity

Since (\sec x=\frac{1}{\cos x}), the equation (\sec\frac{3\theta}{2} = 2) can be rewritten as (\frac{1}{\cos\frac{3\theta}{2}}=2), which implies (\cos\frac{3\theta}{2}=\frac{1}{2}).

Step2: Solve for (\frac{3\theta}{2})

We know that if (\cos x = \frac{1}{2}), then (x = 2k\pi\pm\frac{\pi}{3},k\in\mathbb{Z}). So, (\frac{3\theta}{2}=2k\pi\pm\frac{\pi}{3}).

Step3: Solve for (\theta)

Multiply both sides of (\frac{3\theta}{2}=2k\pi\pm\frac{\pi}{3}) by (\frac{2}{3}) to get (\theta=\frac{4k\pi}{3}\pm\frac{2\pi}{9}).

Step4: Find solutions in the interval (0\leq\theta<2\pi)

When (k = 0):
- (\theta=\frac{2\pi}{9}) (from (\theta=\frac{4k\pi}{3}+\frac{2\pi}{9}))
- (\theta =-\frac{2\pi}{9}+ \frac{4\pi}{3}=\frac{- 2\pi + 12\pi}{9}=\frac{10\pi}{9}) (from (\theta=\frac{4k\pi}{3}-\frac{2\pi}{9}))
When (k = 1):
- (\theta=\frac{4\pi}{3}+\frac{2\pi}{9}=\frac{12\pi + 2\pi}{9}=\frac{14\pi}{9})
- (\theta=\frac{4\pi}{3}-\frac{2\pi}{9}=\frac{12\pi-2\pi}{9}=\frac{10\pi}{9}) (already counted)
When (k = 2):
- (\theta=\frac{8\pi}{3}+\frac{2\pi}{9}=\frac{24\pi + 2\pi}{9}=\frac{26\pi}{9}>2\pi)
- (\theta=\frac{8\pi}{3}-\frac{2\pi}{9}=\frac{24\pi-2\pi}{9}=\frac{22\pi}{9}>2\pi)

Answer:

A. The solution set is (\left{\frac{2\pi}{9},\frac{10\pi}{9},\frac{14\pi}{9}\right})