find all solutions to the equation in the interval 0,2π) sin 2x = 2 cos x select the correct choice and, if…

find all solutions to the equation in the interval 0,2π) sin 2x = 2 cos x select the correct choice and, if necessary, fill in the answer box to complete your choice. a. the solution(s) is/are (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) b. there are no solutions

find all solutions to the equation in the interval 0,2π) sin 2x = 2 cos x select the correct choice and, if necessary, fill in the answer box to complete your choice. a. the solution(s) is/are (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) b. there are no solutions

Answer

Explanation:

Step1: Use double - angle formula

Recall $\sin2x = 2\sin x\cos x$. The given equation $\sin2x = 2\cos x$ becomes $2\sin x\cos x=2\cos x$.

Step2: Rearrange the equation

Subtract $2\cos x$ from both sides: $2\sin x\cos x - 2\cos x=0$. Factor out $2\cos x$: $2\cos x(\sin x - 1)=0$.

Step3: Set each factor equal to zero

Set $2\cos x = 0$ and $\sin x - 1=0$. For $2\cos x = 0$, then $\cos x = 0$. In the interval $[0,2\pi)$, $x=\frac{\pi}{2},\frac{3\pi}{2}$. For $\sin x - 1=0$, then $\sin x = 1$. In the interval $[0,2\pi)$, $x = \frac{\pi}{2}$.

Answer:

$x=\frac{\pi}{2},\frac{3\pi}{2}$