In a right triangle ABC with a hypotenuse AB cosB = 4cosA. Find the AC: BC ratio.

We use the definition of the cosine of an angle in a right-angled triangle, we write this for angle B (adjacent leg – BC) and angle A (adjacent leg – AC):
Cos B = BC / AB
Cos A = AC / AB.
By condition, the cosine of angle B is 4 times greater than the cosine of angle A. We write the equality:
BC / AB = 4 * AC / AB.
BC * AB = AB * 4AC
ВС = 4АС, we divide both parts into 4 and into ВС.
¼ = AC / BC
Answer: AC: BC = ¼.