Final answer:
To find the smallest positive and largest negative co-terminal angles for a point (-5, -3), you first determine the principal angle in the third quadrant and then add or subtract 360° to find the co-terminal angles. The correct answer is 237 degrees for the smallest positive angle and -123 degrees for the largest negative angle.
Step-by-step explanation:
To find the smallest positive and largest negative co-terminal angles for a point (-5, -3) on a terminal arm in standard position, you first need to determine the angle that the point makes with the positive x-axis. As the point is in the third quadrant, we know the angle must be greater than 180 degrees but less than 270 degrees. Using trigonometric ratios, specifically arctan, we get the reference angle. The arctan of 3/5 gives us a reference angle. However, since we are in the third quadrant, we must add 180 degrees to find the principal angle θ, which is θ = 180 + arctan(3/5).
We use the fact that co-terminal angles are angles that differ by full rotations (360°) to find the smallest positive co-terminal angle and the largest negative co-terminal angle. Adding 360° to the principal angle will give us the smallest positive co-terminal angle. Conversely, subtracting 360° will provide the largest negative co-terminal angle.
The correct answer from the given options is therefore D) Smallest positive: 237 degrees; Largest negative: -123 degrees.