In a right-angled triangle, the lengths of the sides form an arithmetic progression. The cosine of its larger acute angle is?

Let the first leg be x, then the second leg is (x + d), and the hypotenuse is (x + 2 * d).
To compose the equation, we use the Pythagorean theorem:
x² + (x + d) ² = (x + 2 * d) ²;
x² – 2 * d * x – 3 * d² = 0;
D = 4 * d² + 4 * 3 * d² = 16 * d²;
√D = 4 * d;
x1 = (2 * d – 4 * d) / 2 = – d – discard the negative root;
x2 = (2 * d + 4 * d) / 2 = 3 * d – the smaller leg of the triangle.

Let’s find what the hypotenuse of the triangle is equal to:
3 * d + 2 * d = 5 * d.

We calculate the value of the cosine as the ratio of the adjacent leg to the hypotenuse:
cos A = (3 * d) / (5 * d) = 3/5 = 0.6.