Find all solutions to the equation $10 \cos (x + 2) = 2$ for $0 \leq x \leq 2 π$ .
Enter your answers in increasing order rounded to three decimal places.

Answer

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Answer

The final solution is $3.014$ .

Steps

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 $0 \leq x \leq 2 π$ , we know that $0 \leq x + 2 \leq 2 π + 2$ , which means $- 2 \leq x \leq 2 π$ .

Step 5 ： $\cos^{- 1} (0.2)$ has two solutions in the range $- 2 \leq x \leq 2 π$ , 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 $0 \leq x \leq 2 π$ , so we discard it.

Step 8 ：The final solution is $3.014$ .