The legs of a right triangle are √15 and 1. Find the sine of the smallest angle of this triangle.

Because the triangle is rectangular, we can apply the Pythagorean theorem:
AB ^ 2 = BC ^ 2 + CA ^ 2
AB ^ 2 = (√15) ^ 2 + 1 ^ 2
AB ^ 2 = 15 + 1 = 16
AB = 4
The smallest angle lies opposite the smallest side (by the theorem on the ratio of sides and angles).
Then the smallest angle is ∠ABC (since 1 <√15).
sin (∠ABC) = AC / AB = 1/4 = 0.25
Answer: the sine of the smallest angle is 0.25