The problem is asking for two angles in the range of 0 to 360 degrees for which the sine of the angle equals -2/3.
To find these angles, we'll first calculate the arcsine of -2/3. The arcsine function, also known as the inverse sine function, will give us an angle whose sine is -2/3. However, due to the nature of the arcsine function, its output will give us an angle only in the first or fourth quadrant.
Since the sine function is positive in the first quadrant and negative in the fourth, to find the angles in the second or third quadrant (where sine is negative), we must transpose the output. Hence, we subtract this output angle from 180 degrees for the second quadrant and from 360 degrees for the third quadrant.
Taking the arcsine of -2/3, we get a negative angle. Converting this angle to degrees, we get an angle in the fourth quadrant. Transposing this angle to the third quadrant, we subtract it from 180 degrees to get its counterpart in the third quadrant which comes out to be approximately around 221.81 degrees. Similarly, transposing this angle to the fourth quadrant, we subtract it from 360 degrees to obtain approximately 318.19 degrees.
Thus, the two angles in the domain 0 to 360 degrees for which the sine function equals -2/3 are approximately 221.81 degrees and 318.19 degrees. This implies that the answer is 'D. None of the above' since none of the alternatives A, B, and C match with these two calculated angles.