In triangle ABC, angle C = 90 degrees. AB = 5, tgA = 7/24 Find AC.

The trigonometric identity is known: tg ^ 2 (x) + 1 = 1 / cos ^ 2 (x). Transforming it, we find
cos ^ 2 (A) = 1 / (tg ^ 2 (A) +1) = 1 / (49/576 + 1) = 1 / (49/576 + 576/576) = 576/625.
Since there are two acute angles in a right-angled triangle, their cosines are positive, so cosA = √ (576/625) = 24/25.
The leg AC is adjacent to the angle A, the ratio of the adjacent leg to the hypotenuse is the cosine of the angle, which means:
AC / AB = cosA;
AC = AB * cosA = 5 * 24/25 = 24/5 = 4.8.