<p>I have an ellipse centered at (0,0) and the bounding rectangle is x = [-5,5], y = [-6,6]. The ellipse intersects the rectangle at (-5,3),(-2.5,6),(2.5,-6),and (5,-3)</p>

<p>I know nothing else about the ellipse, but the only thing I need to know is what angle the major axis is rotated at.</p>

<p>seems like the answer must be really simple but I'm just not seeing it... thanks for the help!</p>

The gradient of the ellipse is identical to the gradient of the intersects with the bounding rectangle along one side of the ellipse.  In your case, that's the line from (-2.5,6) to (5,-3), the top side of your ellipse.  That line has a vertical drop of 9 and a horizontal run of 7.5.
So we end up with the following right-angled triangle.
```
```
(-2.5,6)
  *-----
  |\x
  | \
  |  \
9 |   \
  |    \
  |    x\
  +------* (5,-3)
    7.5
```
```
The angle we're looking for is x which is the same in both locations.
We can calculate it as:
```
```
-1
tan   (9/7.5)
```
```
which gives us an angle of -50.19 degrees