Question

How do I find the angle of rotation of `153x^2 - 192xy + 97y^2 - 30x -40y - 200 = 0`, given `cot 2theta = (A - C)/B`?

Answer 1

`theta = 1/2cot^-1(-56/192) ~~ 53.13^@` (Using a calculator.)

Explanation:

`cot 2theta = (153-97)/(-192) = -56/192`

So `2theta = cot^-1(-56/192)` and

`theta = 1/2cot^-1(-56/192)` Get out a calculator or trig tables. That is not a special angle we know.

But wait, there's more.

Perhaps we are really interested in eliminating the `xy` term from the equation. In which case, we don't really need to know `theta`, we just need the sine and cosine of `theta`. Those we can get without trig tables or calculators.

`-56/192 = -28/96 = -7/24` AHA! (or perhaps not)
Even if we don't immediately see it, that is a `7, 24, 25` right triangle.
Or use trigonometry: if the cotangent of an angle is `-7/24`, find the cosine of that angle.

If `cot 2theta = -7/24`, the `cos 2 theta = -7/25`.

Now we can use half-angle formulas to get `sin theta` and cos theta#

And then we can do the substitution to eliminate the `xy` term.