In a right-angled triangle ABC, the hypotenuse AB is 6 cm, and the leg BC is 3 cm.

In a right-angled triangle ABC, the hypotenuse AB is 6 cm, and the leg BC is 3 cm. Find the second leg and acute angles of the triangle.

A right-angled triangle is a triangle in which one of the corners is a straight line.

To calculate the angle ∠B, we apply the cosine theorem. The cosine of an acute angle in a right triangle is the ratio of the adjacent leg to the hypotenuse:

cos B = BC / AB;

cos B = 3/6 = 1/24

1/2 = cos 60º.

To calculate the value of the angle ∠A, we apply the theorem of sines, according to which the sine of an acute angle of a right-angled triangle is equal to the ratio of the opposite leg to the hypotenuse:

sin A = BC / AB;

sin A = 3/6 = 1/2;

1/2 = sin 30º.

To calculate the AC leg, we use the Pythagorean theorem, according to which the square of the hypotenuse is equal to the sum of the squares of the legs:

AB ^ 2 = BC ^ 2 + AC ^ 2;

AC ^ 2 = AB ^ 2 – BC ^ 2;

AC ^ 2 = 6 ^ 2 – 3 ^ 2 = 36 – 9 = 25;

AC = √25 = 5 cm.

Answer: angle ∠B is 60º, angle ∠A is 30º, leg AC is 5 cm