Solve the equation on the interval $[0,2 \pi)$.
\[
2 \sin 2 x-1=0
\]

Answer

Expert–verified

Hide Steps

Answer

Final Answer: The solutions to the equation $2 \sin 2x - 1 = 0$ in the interval $[0,2 \pi)$ are $\boxed{0.2618}$ and $\boxed{1.3090}$.

Steps

Step 1 ：Given the equation $2 \sin 2x - 1 = 0$.

Step 2 ：Isolate the trigonometric function: $2 \sin 2x = 1$ which simplifies to $\sin 2x = 0.5$.

Step 3 ：The solutions to $\sin 2x = 0.5$ in the interval $[0,2 \pi)$ are approximately 0.2618 and 1.3090.

Step 4 ：Final Answer: The solutions to the equation $2 \sin 2x - 1 = 0$ in the interval $[0,2 \pi)$ are $\boxed{0.2618}$ and $\boxed{1.3090}$.