TITLE: If sum of triangle angles is $180$ degrees, how $\sin(270)$ is possible?
QUESTION [11 upvotes]: I'm not new to trigonometry, but this question always bothers me.
As it is in Wolfram MathWorld-
$$
\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}
$$
We know that the sum of the angles in a triangle is 180 degrees, which means each angle is less than 180 degrees, so how, for example, $\sin 270$  is possible?

Edit
After Joe posted his answer, I went back to MathWorld and read it more carefully. I noticed what Brian Drake mentioned in his answer. MathWorld has already explained both definitions. I kept the question because of the brilliant answers below.

REPLY [2 votes]: I was very surprised to see this question: MathWorld is not the sort of site that would gloss over this, so why can’t we find the answer there?
In fact, MathWorld acknowledges the problem mentioned in the question (emphasis added):

The common schoolbook definition of the sine of an angle $\theta$ in a right triangle (which is equivalent to the definition just given) is …
$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}.\tag{1}$

Here is the “definition just given”:

Let $\theta$ be an angle measured counterclockwise from the $x$-axis along an arc of the unit circle. Then $\sin \theta$ is the vertical coordinate of the arc endpoint, as illustrated in the left figure above.

To be fair, this text and the figure given in MathWorld don’t do a great job of explaining this definition to people not already familiar with it. Hopefully some of the other answers and comments here will explain it in a way you can understand.
But I would suggest that you try to read sites like MathWorld more carefully before turning elsewhere.