The values given for tan α and sin β satisfy the conditions set based on their respective quadrants in the unit circle. Therefore, the given values are the exact values for these trigonometric functions under the specified conditions.
In order to solve the problem, we need to use the given conditions and trigonometric identities. For the first condition
tan α = -12/5
and π/2 < α < pi which indicates that α is in the second quadrant where tan is negative as given. So, this condition is satisfied. Now, for the second condition
sin β = √3/2
, and 0< β < π/2 which tells us that β is in the first quadrant where sin is positive. So, this condition is also satisfied. It's worth mentioning here that in mathematics, angles are taken counterclockwise from the x-axis. So, the quadrant in which the angle lies is important in determining the sign (positive or negative) of trigonometric functions. In conclusion, the exact values of tan α and sin β under the given conditions have been found using their respective quadrant rules.