In an isosceles triangle ABC with base AC, the side AB is 25 and the height to the base is 20. Find the cosine of the angle.

First way.
Since ВС is the height of the ABC triangle, then the ABН triangle is rectangular, in which we determine the length of the AН leg using the Pythagorean theorem.
AH ^ 2 = AB ^ 2 – BH ^ 2 = 625 – 400 – 225.
AH = 15 cm.
Then CosBAC = AH / AB = 15/25 = 3/5.

Second way.
In the right-angled triangle AСН, we define the sine of the angle ВAН.
SinBAH = BH / AB = 20/25 = 4/5.
Determine the cosine of the angle ВAН.
Sin2BAH + Cos2BAH = 1
Cos2BAH = 1 – Sin2BAH = 1 – (4/5) 2 = 1 – 16/25 = 9/25.
CosBAH = 3/5.
Answer: The cosine of the angle is 3/5.