Questions: Find all real solutions to the equation -10 sin x = 5 sqrt(3) between [0°, 360°). A. 150°, 210° B. 240°, 300° C. 60°, 120° D. -60°, -120°

Solution Steps

The given equation is

\[ -10 \sin x = 5 \sqrt{3} \]

First, divide both sides by \(-10\) to isolate \(\sin x\):

\[ \sin x = -\frac{5 \sqrt{3}}{10} = -\frac{\sqrt{3}}{2} \]

The reference angle for \(\sin x = \frac{\sqrt{3}}{2}\) is \(60^\circ\). Since \(\sin x\) is negative, we need to find angles in the third and fourth quadrants.

In the third quadrant, the angle is:

\[ 180^\circ + 60^\circ = 240^\circ \]

In the fourth quadrant, the angle is:

\[ 360^\circ - 60^\circ = 300^\circ \]

Final Answer