Find all solutions to the equation $10 \cos (x+2)=2$ for $0 \leq x \leq 2 \pi$.
Enter your answers in increasing order rounded to three decimal places.
Answer
The final solution is \(\boxed{3.014}\).
Step 1 ：We have that \(10 \cos (x+2)=2\).
Step 2 ：Divide both sides by 10, we get \(\cos (x+2)=0.2\).
Step 3 ：Take the inverse cosine of both sides, we get \(x+2=\cos^{-1}(0.2)\).
Step 4 ：Since \(0 \leq x \leq 2 \pi\), we know that \(0 \leq x+2 \leq 2 \pi + 2\), which means \(-2 \leq x \leq 2 \pi\).
Step 5 ：\(\cos^{-1}(0.2)\) has two solutions in the range \(-2 \leq x \leq 2 \pi\), which are approximately 1.369 and 5.014.
Step 6 ：Subtract 2 from both solutions, we get \(x \approx -0.631\) and \(x \approx 3.014\).
Step 7 ：But \(-0.631\) is not in the range \(0 \leq x \leq 2 \pi\), so we discard it.
Step 8 ：The final solution is \(\boxed{3.014}\).
