To find [tex]\(\theta\)[/tex] when [tex]\(0^\circ < \theta < 360^\circ\)[/tex] and [tex]\(\sin \theta = -\frac{\sqrt{3}}{2}\)[/tex] with [tex]\(\theta\)[/tex] in the fourth quadrant (QIV), follow these steps:
1. Determine the reference angle:
- The reference angle is the angle in the first quadrant that has the same sine value ignoring the sign. For [tex]\(\sin \theta = \frac{\sqrt{3}}{2}\)[/tex], the reference angle is [tex]\(60^\circ\)[/tex].
2. Place the angle in the correct quadrant:
- Since we know [tex]\(\sin \theta = -\frac{\sqrt{3}}{2}\)[/tex], and sine is negative in the fourth quadrant, we will use the reference angle to find [tex]\(\theta\)[/tex] in QIV.
3. Calculate the angle in the fourth quadrant:
- In the fourth quadrant, the angle can be found using [tex]\(360^\circ\)[/tex] minus the reference angle.
- Thus, [tex]\(\theta = 360^\circ - 60^\circ\)[/tex].
- So, [tex]\(\theta = 300^\circ\)[/tex].
Therefore, [tex]\(\theta = 300^\circ\)[/tex] and the reference angle is [tex]\(60^\circ\)[/tex].
