To solve the problem of finding [tex]\(\theta\)[/tex] to the nearest tenth of a degree given [tex]\(\sin \theta = -0.5446\)[/tex] and [tex]\(\theta\)[/tex] in the third quadrant (QIII), we'll follow these steps:
1. Find the reference angle:
The reference angle associated with [tex]\(\sin \theta = -0.5446\)[/tex] is the angle whose sine value is 0.5446.
Using a calculator to find the inverse sine (arcsin) of 0.5446, we get:
[tex]\[
\theta_{\text{reference}} \approx -32.9973^\circ
\][/tex]
2. Adjust for the third quadrant:
Angles in the third quadrant range from [tex]\(180^\circ\)[/tex] to [tex]\(270^\circ\)[/tex]. Since our initial result is negative and [tex]\(\sin \theta\)[/tex] is negative in QIII, we need to find the equivalent angle in QIII.
The equivalent angle in QIII can be determined by using the fact that [tex]\(\sin(180^\circ + \theta_{\text{reference}})\)[/tex] will match [tex]\(\sin \theta\)[/tex]:
[tex]\[
\theta = 180^\circ - (-32.9973^\circ)
\][/tex]
3. Calculate the angle in QIII:
[tex]\[
\theta = 180^\circ + 32.9973^\circ = 212.9973^\circ
\][/tex]
4. Round to the nearest tenth:
[tex]\[
\theta \approx 213.0^\circ
\][/tex]
Thus, the angle [tex]\(\theta\)[/tex] in the third quadrant where [tex]\(0^\circ < \theta < 360^\circ\)[/tex] and [tex]\(\sin \theta = -0.5446\)[/tex] is approximately [tex]\(213.0^\circ\)[/tex].
