Question

How do you solve a triangle given A=58 degrees, a=11.4, b=12.8?

Answer 1

Case 1 : `color(blue)(hat A -= 58, hat B = 72.2, hat C = 49.8, a = 11.4, b = 12.8, c = 10.27`

Case 2 `color(green)(hat A = 58, hat B = 106.8, hat C = 14.2, a = 11.4, b = 12.8, c = 3.3`

Explanation:

`a = 11.4, b = 12.8, hat A = 58^@`

Applying Law of Sines,

`a / sin A= b / sin B = c / sin C`

`sin B = (b * sin A) / a`

`sin B = (12.8 * sin 58)/ 11.4 = 0.9522`

`hat B = sin ^ -1 0.9522 = 72.2^@`

Also sin B = sin (180 - B)#

Hence `hat B = 180-72.2 = 107.8^@`

`hat C = 180 - 58 - 72.2 = 49.8^@`

Also `hat C = 180 - 58 - 107.8 = 14.2^@`

`c = (a * sin C) / sin A = (11.4 * sin 49.8) / sin 58`

`c = 10.27`

Also `c = (11.4*sin 14.2)/sin 58 = 3.3`

Measurements of triangle are

Case 1 : `color(blue)(hat A -= 58, hat B = 72.2, hat C = 49.8, a = 11.4, b = 12.8, c = 10.27`

Case 2 `color(green)(hat A = 58, hat B = 106.8, hat C = 14.2, a = 11.4, b = 12.8, c = 3.3`