Question

Solve the triangle​ ABC, if the triangle exists. B=33 degrees 42' a=37.1 b=27.9?

Answer 1

`color(blue)("Case 1 : :hat A = 47.54^@, hat C = 98.76^@, c = 49.7, A_t = 511.51`

`color(crimson)("Case 2 : " hat A = 132.46^@, hat C = 13.84^@, c = 14.26, A_t = 123.8`

Explanation:

`hat B = 33^@ 42', a = 37.1, b = 27.9`

As per Law of Sines, `a / sin A = b / sin B`

`sin A = (sin B * a) / b = (sin (33.7)^@ * 37.1) / 27.9`

`sin A = 0.7378`

`hat A = sin ^-1 0.7378 = 47.54^@ " or " 132.46^@`

`hat C = 180 - 33.7 - 47.54 = 98.76^@ " or" 180 - 33.7 - 132.46 = 13.84^@`

`"Case 1 : " c = (sin C * b) / sin B = (sin 98.76 * 27.9) / sin 33.7 = 49.7`

`"Area " = A_t = (1/2) a b sin C = (1/2) * 37.1 * 27.9 * sin 98.76 = 511.51`

`"Case 2 : " c = (sin C * b) / sin B = (sin 13.84 * 27.9) / sin 33.7 = 14.26`

`"Area " = A_t = (1/2) a b sin C = (1/2) * 37.1 * 27.9 * sin 13.84 = 123.8`