Question
Find the measure of θ given the following information.(2 points) If sin θ = −12 and θ is in Quadrant 3, then θ= degrees or radians.
Answers
GPT-4o mini
It seems there’s a mistake in the value of \(\sin \theta\) that you provided, because the sine function only outputs values in the range \([-1, 1]\). Thus, \(\sin \theta = -12\) is not possible.
If you meant \(\sin \theta = -\frac{1}{2}\) (as an example of a valid sine value), we could find \(\theta\) in quadrant 3.
In quadrant 3, sine is negative. The reference angle for \(\sin^{-1}(-\frac{1}{2})\) is \(\frac{\pi}{6}\) or \(30^\circ\). However, to find the angle in the third quadrant:
\[
\theta = \pi + \text{reference angle}
\]
So,
\[
\theta = \pi + \frac{\pi}{6} = \frac{6\pi}{6} + \frac{\pi}{6} = \frac{7\pi}{6} \text{ radians}
\]
In degrees, this would be:
\[
\theta = 180^\circ + 30^\circ = 210^\circ.
\]
If you provide the correct value for \(\sin \theta\), I can help you find the angle accordingly.
If you meant \(\sin \theta = -\frac{1}{2}\) (as an example of a valid sine value), we could find \(\theta\) in quadrant 3.
In quadrant 3, sine is negative. The reference angle for \(\sin^{-1}(-\frac{1}{2})\) is \(\frac{\pi}{6}\) or \(30^\circ\). However, to find the angle in the third quadrant:
\[
\theta = \pi + \text{reference angle}
\]
So,
\[
\theta = \pi + \frac{\pi}{6} = \frac{6\pi}{6} + \frac{\pi}{6} = \frac{7\pi}{6} \text{ radians}
\]
In degrees, this would be:
\[
\theta = 180^\circ + 30^\circ = 210^\circ.
\]
If you provide the correct value for \(\sin \theta\), I can help you find the angle accordingly.