Answer:
We can use the definition of tangent to find the value of \tan\theta. Tangent is defined as the ratio of the opposite side to the adjacent side in a right triangle.
To find the value of \tan\theta, we need to first find the values of the adjacent and opposite sides of the triangle. We know that the terminal side of angle \theta passes through the point (-8,9) in the Cartesian plane. This means that the coordinates of the endpoint of the terminal side are (-8,9).
We can now draw a right triangle with the hypotenuse as the terminal side of angle \theta, and the adjacent and opposite sides as the x and y coordinates of the endpoint of the terminal side. We can use the Pythagorean theorem to find the length of the hypotenuse.
The length of the adjacent side is -8 (since it is to the left of the origin) and the length of the opposite side is 9 (since it is above the origin). Therefore, we have:
adjacent = -8
opposite = 9
hypotenuse = \sqrt{(-8)^2 + 9^2} = \sqrt{64 + 81} = \sqrt{145}
Now we can use the definition of tangent to find the value of \tan\theta:
\tan\theta = \frac{opposite}{adjacent} = \frac{9}{-8} = -\frac{9}{8}
Therefore, the exact value of \tan\theta is -\frac{9}{8} in simplest radical form.