In a right-angled triangle ABC, the leg is AC = 8, and the height CH dropped to the hypotenuse is 2√15. Find sin∠ABC.

Consider the triangles ABC and ACH.
∠AHC = ∠ACB (as these are right angles).
∠A is common.
Therefore, by the theorem on the sum of the angles of a triangle, ∠ACH = ∠ABC
Then sin∠ACH = sin∠ABC.
Now consider the triangle ACH.
By the Pythagorean theorem:
AC2 = CH2 + AH2
8 ^ 2 = (2√15) ^ 2 + AH ^ 2
64 = 4 * 15 + AH ^ 2
AH ^ 2 = 64-60
AH ^ 2 = 4
AH = 2
sin∠ACH = AH / AC (by definition)
sin∠ACH = 2/8 = 1/4 = 0.25
As deduced above:
sin∠ABC = sin∠ACH = 0.25
Answer: sin∠ABC = 0.25