If the base is 60 and the base angle is 45, what is the length of the two sides?

Both base angles are 45°. Therefore the third angle is 90°. Drop an altitude from the 90° angle to the base. The altitude bisects the 90° angle. It also bisects the base and is perpendicular to it. The altitude forms two smaller isosceles right triangles, each of which has two 45° angles and two sides with lengths of 30 (half the base). Thus, each 45° angle in each smaller right triangle has an opposite side and an adjacent side of length 30 and a hypotenuse of x (the length you're trying to find). The sine (and cosine) of each 45° angle is 0.707. Therefore, 0.707 = 30 / x. x = 30 / 0.707 = 42.4. That's the length of each of the two equal sides of the big triangle.