In a right-angled triangle ABC, angle C = 90 degrees, angle A = 30 degrees, AB = 10 cm. Find AC, BC and angle B.

1) Find angle B.

In a right-angled triangle, the sum of the angles at the base is 90 °.

A + B = 90 °;

B = 90 ° – A;

B = 90 ° – 30 ° = 60 °.

2) Find the length of the BC side.

The leg lying opposite an angle of 30 ° is equal to half the hypotenuse. The leg BC lies opposite an angle of 30 °, hypotenuse AB = 10 cm.

BC = AB: 2;

BC = 10: 2;

BC = 5 cm.

3) Find AC through the sine of angle B.

Sine of angle B is the ratio of the opposite leg of the AC to the hypotenuse AB:

Sin B = AC: AB;

AC = AB × sin B.

AC = 10 × sin 60 °;

AC = 10 × √3: 2;

AC = 5√3 cm.