In an acute-angled triangle ABC, the height AH is 20√3 and the side AB is 40. Find cosB.

АН – height, which means that it is perpendicular to the BC side, we get:

right angle СНА = angle АНВ = 90;

Then triangle AНВ is right-angled.

Leg: AН and HB. Hypotenuse AB.

Using the Pythagorean theorem, we find the value of the unknown leg HB.

HB ^ 2 = AB ^ 2 – AH ^ 2;

HB ^ 2 = 40 ^ 2 – (20 √ 3) ^ 2 = 1600 – 400 * 3 = 1600 – 1200 = 400;

HB ^ 2 = √ 400 = 20;

by the definition of trigonometric functions, the Cos of the angle is calculated as the ratio of the adjacent leg to the hypotenuse:

Cos B = HB / AB

Cos B = 20/40 = 1/2 = 0.5.