Find the ratio of the heights of the triangle ABC, dropped from the vertices A and B, respectively, if cosα = 1/5, sin β = 1/2.

Since AH is the height, the triangle ABH is rectangular in which AH = AB * SinABH = AB / 2.

Determine the sine of the angle BAC. Sin2BAC = 1 – Cos2BAC = 1 – 1/25 = 24/25.

SinBAC = 2 * √6 / 5.

ВK – height, then triangle ABK is rectangular in which BK = AB * SinBAC = AB * 2 * √6 / 5.

Then ВK / AН = (AB * 2 * √6 / 5) / (AB / 2) = 4 * √6 / 5.

Answer: The ratio of the legs is 4 * √6 / 5.