The prior answers have all used calculus. I'm going to post an answer using only trig.
The following diagram from Wikipedia's Trig Page is helpful.

However, that diagram also has a fault--the picture is very cluttered. :) Thus, I've redrawn it for you, labeling the components important for this problem:

Note that $\csc\theta$ returns the distance from the origin to the y-intercept of the tangent line, and $\sec\theta$ returns the distance from the origin to the x-intercept of the tangent line.
Let $m$ represent slope: $m=\frac{\Delta y}{\Delta x}=\frac{\csc\theta}{\sec\theta}$ Simplifying, this gives: $m=\cot\theta$
Now, for your particular point, you will run into some difficulty: $m=\cot\pi=undefined$ To interpret this, note that $\cot\pi$ is undefined because you are dividing by zero. This means that you had some change in the $y$ direction, but none in the $x$ direction. The only type of line like this is a vertical one. :)
Let me know if this needs more explaining--I'll edit the answer to make it clearer.