A right triangle has a hypotenuse of length 40 and an angle of 25 degrees with a side opposite this angle of length 16. A second right triangle also has an angle of 25 degree with a hypotenuse of length 10. Determine the length of the side opposite the 25 degree angle on the second triangle
In the first triangle, let $x$ be the length of the side opposite the 25 degree angle. Since the hypotenuse has length 40 and the side opposite the 25 degree angle has length 16, by the Pythagorean Theorem the other leg must have length $\sqrt{40^2 - 16^2} = 24\sqrt{3}$. Then by the definition of sine, we have $\sin 25 = \frac{16}{40} = \frac{2}{5}$, so $\frac{x}{10} = \frac{2}{5}$ and $x = \boxed{4}$.
