What's the speed?

Warning: trigonometry ahead

Here's another mathish problem for your enjoyment. There's no trick here, it's just a tough problem.

There are four points, A, B, C, and D. You wish to travel from point A to point C. You travel in a vehicle that, when it's engaged, only goes at exactly one speed; no more, no less.

You know that angle BCD = 90 degrees. You also know that the length of segment DC = 10 miles, and the length of BC also equals 10 miles.

There are three paths you can take to get from A to C:
Going from A straight to C takes 30 minutes.
Going from A to B to C takes 35 minutes.
Going from A to D to C takes 40 minutes.

See the following illustration (it's clickable):



For convenience, I've defined angle ACD as theta and angle ACB as 90-theta.

Find the speed of your vehicle.


I'll post a solution in a week or so.

**UPDATE**
Hong Kim busted up some trig and got the solution. Here's my solution to the problem. Hong, is this similar to the approach you took?