Simplify the equation by using suitable identities if needed.

Determine the reference angle, and use the value of the trigonometric ratio without taking into consideration the signs.

Find the angles in the quadrants that correspond to the signs of the trigonometric ratio and range.

Write the solutions obtained.

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

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

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

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