solve the equation on the interval (0leq\theta<2pi).\n(sec\frac{5\theta}{4}=2)\nwhat are the solutions to…

solve the equation on the interval (0leq\theta<2pi).\n(sec\frac{5\theta}{4}=2)\nwhat are the solutions to (sec\frac{5\theta}{4}=2) in the interval (0leq\theta<2pi)? select the correct cho\no a. the solution set is ({}).\n(simplify your answer. type an exact answer, using (pi) as needed. type your \nanswers as needed.)\no b. there is no solution.

solve the equation on the interval (0leq\theta<2pi).\n(sec\frac{5\theta}{4}=2)\nwhat are the solutions to (sec\frac{5\theta}{4}=2) in the interval (0leq\theta<2pi)? select the correct cho\no a. the solution set is ({}).\n(simplify your answer. type an exact answer, using (pi) as needed. type your \nanswers as needed.)\no b. there is no solution.

Answer

Explanation:

Step1: Use the reciprocal identity

Since (\sec x=\frac{1}{\cos x}), the equation (\sec\frac{5\theta}{4} = 2) can be rewritten as (\frac{1}{\cos\frac{5\theta}{4}}=2), which implies (\cos\frac{5\theta}{4}=\frac{1}{2}).

Step2: Find the general solutions for (\cos u=\frac{1}{2})

The general solutions for (\cos u=\frac{1}{2}) are (u = 2k\pi\pm\frac{\pi}{3}), where (k\in\mathbb{Z}). Substituting (u = \frac{5\theta}{4}), we get (\frac{5\theta}{4}=2k\pi\pm\frac{\pi}{3}).

Step3: Solve for (\theta)

Multiply both sides of (\frac{5\theta}{4}=2k\pi\pm\frac{\pi}{3}) by (\frac{4}{5}). So (\theta=\frac{8k\pi}{5}\pm\frac{4\pi}{15}).

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

  • When (k = 0):
    • (\theta=\frac{4\pi}{15}) (from (\theta=\frac{8k\pi}{5}+\frac{4\pi}{15}))
    • (\theta=\frac{8\pi}{15}) (from (\theta=\frac{8k\pi}{5}-\frac{4\pi}{15}))
  • When (k = 1):
    • (\theta=\frac{8\pi}{5}+\frac{4\pi}{15}=\frac{24\pi + 4\pi}{15}=\frac{28\pi}{15})
    • (\theta=\frac{8\pi}{5}-\frac{4\pi}{15}=\frac{24\pi- 4\pi}{15}=\frac{20\pi}{15}=\frac{4\pi}{3})

Answer:

A. The solution set is (\left{\frac{4\pi}{15},\frac{8\pi}{15},\frac{4\pi}{3},\frac{28\pi}{15}\right})