In a right-angled triangle, the leg is 8cm and the opposite angle is 30 °. Find unknown elements of a triangle?

First, let’s find the value of the second acute angle of this right-angled triangle.

Since the sum of all the angles of any triangle is 180 °, and one of the acute angles of this right-angled triangle is 30 °, the second acute angle is 180 – 30 – 90 = 150 – 90 = 60 °.

Let us denote by b the leg lying opposite to an angle of 60 °, and by c – the hypotenuse.

Using the sine theorem, we find the sides b and c:

b = 8 * sin (60 °) / sin (30 °) = 8 * (√3 / 2) / (1/2) = 8√3 cm;

c = 8 * sin (90 °) / sin (30 °) = 8 * 1 / (1/2) = 16 cm.

Answer: 8√3 cm and 16 cm.