In triangle ABC, angle C is 90 °, angle B is 60 °, CB = 3cm. Identify the speaker.

A triangle in which one of the angles is 90º is called rectangular. The side opposite the right angle is called the hypotenuse, and the other two are called legs.

For the convenience of calculating the length of the AC leg, we find the length of the hypotenuse AB. For this we apply the cosine theorem. The cosine of an acute angle of a right-angled triangle is the ratio of the adjacent leg to the hypotenuse:

cos B = BC / AB;

AB = BC / cos B;

cos 60º = 1/2 = 0.5;

AB = 3 / 0.5 = 6 cm.

To calculate the AC leg, we use the Pythagorean theorem:

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

AC ^ 2 = AB ^ 2 – BC ^ 2;

AC ^ 2 = 6 ^ 2 – 3 ^ 2 = 36 – 9 = 27;

AC = √27 ≈ 5.2 cm.

Answer: the length of the AC leg is 5.2 cm.