Use a calculator to find [tex]$\theta$[/tex] to the nearest tenth of a degree, if [tex]$0^{\circ} \ \textless \ \theta \ \textless \ 360^{\circ}$[/tex] and [tex][tex]$\sin \theta = -0.5446$[/tex][/tex], with [tex]$\theta$[/tex] in Quadrant III.

[tex]\boxed{}[/tex]



Answer :

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].