What's the formula for these special right triangles?

In a 45–45-90 triangle, the measure of the hypotenuse is equal to the measure of a leg multiplied by SQRT(2).

In a 30–60-90 triangle, the measure of the hypotenuse is two times that of the leg opposite the 30o angle. The measure of the other leg is SQRT(3) times that of the leg opposite the 30o angle.

Here are pictures of the relationships @bkcunningham1 just described.
30–60-90
45–45-90

Is this what you were asking for? Or were you wondering about the trig functions associated with those triangles?

If you know the Pythagorean equation, you can work these out on your own. It is convenient to take the hypotenuse as equal to 1. For the 45 degree triangle, both the legs are equal, giving x^2 + x^2 =1, which works out to sqrt(2)/2.

For the 30–60-90 triangle, start with an equillateral triangle and drop an altitutde from the top vertex. This divides the triangle into two 30–60-90 triangles with the 30 degree angle on the top. Since all the sides of the original triangle were equal and the side opposite the 30 degree angle is half the base, the length of that side is ½. You can use the Pythagorean equation x^2 + (½)^2 = 1 to solve for the side opposite the 60 degree angle, which works out to sqrt(3)/2.