Question
Find the measure of 0 given the following information.
If sin 0 = -1/2 and 0 is in Quadrant 3, then 0 = __ degrees or __ radians
If sin 0 = -1/2 and 0 is in Quadrant 3, then 0 = __ degrees or __ radians
Answers
GPT-4o mini
To find the angle \( \theta \) such that \( \sin \theta = -\frac{1}{2} \) in Quadrant 3, we need to determine the reference angle where \( \sin \) is positive.
The sine of an angle equals \( -\frac{1}{2} \) at the reference angles \( 30^\circ \) or \( \frac{\pi}{6} \) because \( \sin 30^\circ = \frac{1}{2} \).
Since we need the angle in Quadrant 3, we find it by subtracting the reference angle from \( 180^\circ \) (or \( \pi \) radians), and then adding \( 180^\circ \) (or \( \pi \) radians) since Quadrant 3 angles are measured from \( 180^\circ \).
Calculating in degrees:
\[
\theta = 180^\circ + 30^\circ = 210^\circ
\]
Calculating in radians:
\[
\theta = \pi + \frac{\pi}{6} = \frac{6\pi}{6} + \frac{\pi}{6} = \frac{7\pi}{6}
\]
Thus, the angles are:
\[
\theta = 210^\circ \text{ or } \frac{7\pi}{6} \text{ radians.}
\]
The sine of an angle equals \( -\frac{1}{2} \) at the reference angles \( 30^\circ \) or \( \frac{\pi}{6} \) because \( \sin 30^\circ = \frac{1}{2} \).
Since we need the angle in Quadrant 3, we find it by subtracting the reference angle from \( 180^\circ \) (or \( \pi \) radians), and then adding \( 180^\circ \) (or \( \pi \) radians) since Quadrant 3 angles are measured from \( 180^\circ \).
Calculating in degrees:
\[
\theta = 180^\circ + 30^\circ = 210^\circ
\]
Calculating in radians:
\[
\theta = \pi + \frac{\pi}{6} = \frac{6\pi}{6} + \frac{\pi}{6} = \frac{7\pi}{6}
\]
Thus, the angles are:
\[
\theta = 210^\circ \text{ or } \frac{7\pi}{6} \text{ radians.}
\]