Question

solve the trigonometric equation for all values $0 \leq x < 2\pi$. $2 \sin x = -1$

Explanation:

Step1: Isolate $\sin x$

Divide both sides by 2:
$$\sin x = \frac{-1}{2}$$

Step2: Find reference angle

The reference angle $\alpha$ satisfies $\sin \alpha = \frac{1}{2}$, so $\alpha = \frac{\pi}{6}$.

Step3: Identify quadrants

Sine is negative in Q3 and Q4.

Step4: Calculate Q3 solution

$$x = \pi + \frac{\pi}{6} = \frac{7\pi}{6}$$

Step5: Calculate Q4 solution

$$x = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6}$$

Answer:

$x=\frac{7\pi}{6}$, $x=\frac{11\pi}{6}$