In triangle ABC, side AB = 5 cm, angle C = 30, angle B = 45. Solve triangle ABC.

1) Find the angle A based on the theorem on the sum of the angles of a triangle: the sum of all the angles of a triangle is 180 degrees – angle A = 180 – 45 – 30 = 105 degrees.

2) From corner A we draw a perpendicular AK to the side BC. Angle AKВ = AKC = 90 degrees. Then, by the theorem on the sum of the angles of a triangle, we find the angles ВAK and СAK:

ВAK = 180 – 45 – 90 = 45 degrees =》 triangle ВAK is isosceles;

СAK = 180 – 30 – 90 = 60 degrees.

3) Find the value of AK. Since ВAK is an isosceles right-angled triangle with a hypotenuse equal to 5 cm, then ВK = AK, sin (ВAK) = cos (ВAK) = 1/2 ^ (- 1/2). Consequently:

AK: 5 = 2 ^ (- 1/2);

AK = 5 * 2 ^ (- 1/2) = 5/2 ^ (1/2) = ВK.

4) Since the leg, which lies opposite an angle of 30 degrees, is equal to half of the hypotenuse, then

AC = 2 * 5/2 ^ (1/2) = 5 * 2 ^ (1/2).

5) Find the CК through cos (30):

СK / AK = 3 ^ (1/2) / 2;

CK = AK * 3 ^ (1/2) / 2 = 5 * 2 ^ (1/2) * 3 ^ (1/2) / 2 = 5 * (3/2) ^ (1/2).

5) Find the side of the aircraft:

BC = ВK + СK = 5 * 2 ^ (- 1/2) + 5 * (3/2) ^ (1/2) = 5 * (3 ^ (1/2) +1) / 2 ^ (1 / 2).

Answer: angle C = 105 degrees, side BC = 5 * (3 ^ (1/2) +1) / 2 ^ (1/2), side AC = 5 * 2 ^ (1/2).