Final answer:
The angle coterminal with 19/6 radians is found by subtracting full rotations of 2π until we are within the range of 0 to 2π. The calculation results in -5π/6, which is within a single rotation and corresponds to option D.
Step-by-step explanation:
The question asks to find the angle that is coterminal with 19/6 radians. Coterminal angles are those that differ by a full rotation of 2π radians or multiple full rotations. Since the student has provided options in terms of π, we can first convert 19/6 to an equivalent angle within one full rotation by subtracting 2π until we are within the range of 0 to 2π radians.
Starting with 19/6, which is greater than 2π (approximately 6.28), we subtract 2π (or approximately 6.28) to bring it within one full rotation:
19/6 - 2π = 19/6 - 12/6 = 7/6 radians, which is still greater than 1π (3.14), so we have to subtract another full rotation to get 7/6 - 2π = -5/6 radians, which is our coterminal angle within the range of 0 to 2π.
After conversion, we find that the equivalent coterminal angle is -5π/6 (option D). It's important to recognize that -5π/6 is the same as adding one full rotation to the angle to get a positive coterminal angle. Therefore, the answer is option D: -5π/6.