Final answer:
The sine of an angle is always positive if the terminal side of the angle lies in the first or second quadrant, where the y-coordinates are positive.
Step-by-step explanation:
When finding the sine of an angle, the result is always positive if the terminal side lies in either the first or second quadrant of the coordinate plane. The reason for this is that sine is defined as the quotient of the opposite side to the hypotenuse in a right triangle, and in these quadrants, the y-coordinate (opposite side) of a point is positive since it lies above the x-axis. When we generalize sine to the unit circle, the y-coordinate of a point on the unit circle corresponds to the sine of the associated angle. In the first quadrant, all angles are between 0 and 90 degrees, and in the second quadrant, they are between 90 and 180 degrees. Within these ranges, the y-coordinates (and therefore the sine values) are positive. This is visually depicted in figures like Figure 5.17, where the y component represents the opposite side and is positive in these quadrants.