One of the legs of a right-angled triangle is 1 full 1/3 times longer than the other. Find the cosines

One of the legs of a right-angled triangle is 1 full 1/3 times longer than the other. Find the cosines of the acute angles of the triangle.

The cosine of an acute angle of a right triangle is the ratio of the adjacent leg to the hypotenuse.

Let’s take one leg of the triangle as x, then the second leg will be equal to 1 1/3 * x = 4x / 3. Let us express the hypotenuse of the triangle through the legs, applying the Pythagorean theorem: The square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs. Let us denote the hypotenuse with.

c ^ 2 = x ^ 2 + (4x / 3) ^ 2 = x ^ 2 + (16x ^ 2) / 9 = (9x ^ 2) / 9 + (16x ^ 2) / 9 = (25x ^ 2) / nine;

c = √ ((25x ^ 2) / 9) = 5x / 3.

Let’s find the cosines of acute angles:

one). cos α = x / (5x / 3) = x * 3 / (5x) = 3/5;

2) cos β = (4x / 3) / (5x / 3) = (4x / 3) * (3 / (5x)) = 4/5.

Answer. 3/5; 4/5.