Find all solutions to the equation $11 \cos x=\frac{1}{\cos x}$ for $0 \leq x \leq 2 \pi$. Enter your answers in increasing order, rounded to three decimal places.
Answer
Final Answer: The solutions to the equation \(11 \cos x=\frac{1}{\cos x}\) for \(0 \leq x \leq 2 \pi\) are \(\boxed{1.265, 1.877, 4.406, 5.019}\).
Step 1 ：Given the equation \(11 \cos x=\frac{1}{\cos x}\). This can be rewritten as \(11 \cos^2 x = 1\).
Step 2 ：Solving for \(\cos x\), we get \(\cos x = \pm \frac{1}{\sqrt{11}}\).
Step 3 ：We need to find the values of \(x\) in the interval \([0, 2\pi]\) that satisfy this equation.
Step 4 ：The solutions to the equation are approximately \(x = 1.265, 1.877, 4.406, 5.019\).
Step 5 ：Final Answer: The solutions to the equation \(11 \cos x=\frac{1}{\cos x}\) for \(0 \leq x \leq 2 \pi\) are \(\boxed{1.265, 1.877, 4.406, 5.019}\).
