Solve the equation \(\text{sin }\theta = -0.5446\) for \(0^{\circ} \leqslant \theta \leqslant 360^{\circ}\).

Reference angle:

\(\begin{aligned} \alpha &= \text{sin}^{-1}(0.5446)\\ \alpha &= 33^{\circ}. \end{aligned}\)

 \(\text{sin }\theta\) is negative, so \(\theta\) is in the quadrant \(\text{III}\) and \(\text{IV}\) for \(0^{\circ} \leqslant \theta \leqslant 360^{\circ}\).

\(\begin{aligned} \theta&=180^\circ+33^\circ \\ &=213^\circ \end{aligned}\) and \(\begin{aligned} \theta&=360^\circ-33^\circ \\ &=327^\circ. \end{aligned}\)

Solve the equation \(\text{sin }\theta = -0.5446\) for \(0^{\circ} \leqslant \theta \leqslant 360^{\circ}\).

Reference angle:

\(\begin{aligned} \alpha &= \text{sin}^{-1}(0.5446)\\ \alpha &= 33^{\circ}. \end{aligned}\)

 \(\text{sin }\theta\) is negative, so \(\theta\) is in the quadrant \(\text{III}\) and \(\text{IV}\) for \(0^{\circ} \leqslant \theta \leqslant 360^{\circ}\).

\(\begin{aligned} \theta&=180^\circ+33^\circ \\ &=213^\circ \end{aligned}\) and \(\begin{aligned} \theta&=360^\circ-33^\circ \\ &=327^\circ. \end{aligned}\)