Final answer:
To find csc A with the given cosine value in Quadrant II, use the Pythagorean identity to find sin A, then take its reciprocal. In this case, csc A is √89/5 with a rational denominator.
Step-by-step explanation:
To find the exact value of csc A when given cos A = -8/√89 and knowing that angle A is in Quadrant II, first remember that the cosecant function is the reciprocal of the sine function, so csc A = 1/sin A. In Quadrant II, the sine function is positive, while cosine is negative. We can use the Pythagorean identity sin2A + cos2A = 1 to find sin A.
Starting with the given cos A = -8/√89, and squaring it, we get cos2A = 64/89. Plugging this into the Pythagorean identity gives us sin2A = 1 - 64/89 = 25/89. Since A is in Quadrant II and sine is positive there, sin A = √25/89 = 5/√89. Therefore, the csc A, which is the reciprocal of sin A, is √89/5 with a rational denominator.