Final answer:
The smallest positive co-terminal angle for the point (-5, -3) is 570.96°, and the largest negative co-terminal angle is -149.04°, considering the point lies in the third quadrant with a principal angle of approximately 210.96°.
Step-by-step explanation:
To find the smallest positive and largest negative co-terminal angle of a principal angle for a point (-5, -3), we first have to determine the principal angle's measure. The point (-5, -3) lies in the third quadrant, which means the principal angle θ is more than 180° but less than 270°. We can use the arctangent function to find the reference angle α, and then subtract α from 180° to get the principal angle θ (since α is in the third quadrant). To find the co-terminal angles, we add or subtract multiples of 360° to the found θ.
The reference angle α can be found using: α = arctan(|-3|/|-5|). Let's assume α is around 30.96°, so θ = 180° + α ≈ 180° + 30.96° = 210.96°. To find the smallest positive co-terminal angle, we add 360° to the principal angle: 210.96° + 360° = 570.96°. To find the largest negative co-terminal angle, we subtract 360°: 210.96° - 360° = -149.04°.