VINCENT WALKER Question 9 of 16 , Step 1 of 1 $6 / 16$ Correct Use trigonometric identities and algebraic methods, as necessary, to solve the following trigonometric equation. Please identify all possible solutions by including all answers in $[0,2 \pi)$ and indicating the remaining answers by using $n$ to represent any integer. Round your answer to four decimal places, if necessary. If there is no solution, indicate "No Solution." \[ 2 \sin ^{2}(x)+2=4 \] Answer How to enter your answer (opens in new window) Enter your answer in radians, as an exact answer when possible. Multiple answers should be separated by commas. Selecting a radio button will replace the entered answer value(s) with the radio button value. If the radio button is not selected, the entered answer is used.
Solution
Step 1 ：The given equation is a trigonometric equation. To solve it, we need to isolate the trigonometric function and then find the possible values of x. The given equation is \(2 \sin ^{2}(x)+2=4\).
Step 2 ：We can start by subtracting 2 from both sides to isolate the trigonometric term. This gives us \(2 \sin ^{2}(x) = 2\).
Step 3 ：Then, we can divide both sides by 2 to get \(\sin ^{2}(x) = 1\).
Step 4 ：Taking the square root of both sides, we get \(\sin(x) = \pm 1\).
Step 5 ：We know that sin(x) equals 1 at \(\frac{\pi}{2}\) and -1 at \(\frac{3\pi}{2}\) in the interval \([0,2 \pi)\).
Step 6 ：Therefore, the solutions to the equation are \(x = \frac{\pi}{2}, \frac{3\pi}{2}\).
Step 7 ：Final Answer: The solutions to the equation are \(x = \boxed{\frac{\pi}{2}, \frac{3\pi}{2}}\).
