Find the exact value of tan theta in simplest radical form if theta is an angle in...

1. Identify the coordinates: The problem states that the terminal side of angle $\theta$ passes through the point $(-7, 9)$. In the Cartesian coordinate system, this point has an x-coordinate of $-7$ and a y-coordinate of $9$. So, we have $x = -7$ and $y = 9$.

1. Recall the definition of tangent: For an angle in standard position, the tangent of the angle, $\tan \theta$, is defined as the ratio of the y-coordinate to the x-coordinate of any point on its terminal side, provided that the x-coordinate is not zero. Mathematically, this is expressed as $\tan \theta = \frac{y}{x}$.

3. Substitute the coordinates into the tangent formula: Using the given coordinates $x = -7$ and $y = 9$, we substitute these values into the formula for tangent:
$$\tan \theta = \frac{9}{-7}$$

1. Simplify the expression: The fraction $\frac{9}{-7}$ can be written as $-\frac{9}{7}$. Since the question asks for the answer in simplest radical form, and there are no radicals in this expression, this is the simplest form.

Therefore, the exact value of $\tan \theta$ is $-\frac{9}{7}$.