Find the sides and angles of a triangle with two sides equal to 17 cm and 36 cm, and the angle between them is 28 degrees.

Given: b = 17 cm, c = 36 cm, α = 28 °.

Find: а -?, Β -?, ϒ -?.

Decision.

Find the side a by the cosine theorem: a ^ 2 = b ^ 2 + c ^ 2 – 2bc * cos α;

a ^ 2 = 17 ^ 2 + 36 ^ 2 – 2 * 17 * 36 * cos 28 ° ≈ 289 + 1296 – 1224 * 0.8829 ≈ 1585 – 1080.6696 ≈ 504.3304;

a ≈ 22.46 (cm).

Let us find the angle β by the theorem of sines: a / (sin α) = b / (sin β);

sin β = (b * sin α) / a;

sin β = (17 * sin 28 °) / 22.46 ≈ (17 * 0.4695) / 22.46 ≈ 7.9815 / 22.46 ≈ 0.3554;

β ≈ 21 °.

Find the angle ϒ by the theorem on the sum of the angles of a triangle: α + β + ϒ = 180 °.

ϒ = 180 ° – (α + β);

ϒ ≈ 180 ° – (28 ° + 21 °) ≈ 180 ° – 49 ° ≈ 131 °.

Answer. a ≈ 22.46 cm; β ≈ 21 °; ϒ ≈ 131 °.