Answer: Let's start by drawing a right triangle with an angle whose sine is equal to -8/9.
We can use the Pythagorean theorem to find the length of the third side:
sin(theta) = opposite / hypotenuse
-8/9 = opposite / 1
opposite = -8/9
adjacent = sqrt(1^2 - (-8/9)^2) = 1/9 * sqrt(65)
So, we have a triangle with an opposite side of -8/9 and an adjacent side of 1/9 * sqrt(65), and a hypotenuse of 1.
Now, we can find the value of cot(theta) using the definition of cotangent:
cot(theta) = adjacent / opposite
= (1/9 * sqrt(65)) / (-8/9)
= -sqrt(65) / 8
Therefore, the exact value of cot[sin^(-1)(-8/9)] is -sqrt(65)/8.
Explanation: