In triangle ABC, angle C is equal to 90th AC = 9 AB = 25. Find sin B.

Given: right-angled triangle ABC;
angle C = 90;
AB = 25;
AC = 9;
Find: sin A

Solution: Consider a right-angled triangle ABC. By the Pythagorean theorem (the square of the hypotenuse is equal to the sum of the squares of the legs):
AC ^ 2 + BC ^ 2 = AB ^ 2 (we express the legs BC ^ 2 from this equality);
BC ^ 2 = AB ^ 2 – AC ^ 2;
BC ^ 2 = 25 ^ 2 – 9 ^ 2;
BC ^ 2 = 625 – 81;

BC ^ 2 = 544;

BC = 23.

The sine of the angle in a right-angled triangle is equal to the ratio of the opposite leg to the hypotenuse. Hence:
sin A = BC / AB;

sin A = 23/25.

Answer: sin A = 23/25.