Question

find all solutions to 2 sin(θ) = -1 on the interval 0 ≤ θ < 2πθ =give your answers as exact values, as a list separated by commas.

Explanation:

Step1: Solve for \(\sin(\theta)\)

Divide both sides of \(2\sin(\theta)= - 1\) by \(2\).
\(\sin(\theta)=-\frac{1}{2}\)

Step2: Find the reference angle

Since \(\sin(\theta)=\frac{1}{2}\) when \(\theta = \frac{\pi}{6}\) (reference angle).

Step3: Determine the quadrants

\(\sin(\theta)<0\) in the third and fourth quadrants.
In the third quadrant: \(\theta=\pi+\frac{\pi}{6}=\frac{7\pi}{6}\)
In the fourth quadrant: \(\theta = 2\pi-\frac{\pi}{6}=\frac{11\pi}{6}\)

Answer:

\(\frac{7\pi}{6},\frac{11\pi}{6}\)