Solve ?ABC............................

Question

Solve ΔABC. (Round your answers for b to one decimal place. Round your answers for β and γ to the nearest 10 minutes. If there is no solution, enter NO SOLUTION.)

α = 23°30',    c = 54.9,    a = 29.2

 

smaller b:

b =	____
β =	__°__'
γ =	__°__'

larger b:

b =	___
β =	__°__'
γ =	__°__'

Answer 1

A=23.5

c = 54.9

a = 29.2

let A=alpha, B=beta, C=gamma

use the law of cosines

a^2 = b^2 + c^2 -2abcosA

29.2^2 = b^2 + 54.9^2 - 2(29.2)b(cos23.5)

b^2 -53.6b -852.6 = 0

use the quadratic formula

b = 53.6/2 + or - (1/2)sqr(53.6^2 -4(-852.6))

b= 66,4 = large b

use the law of sines to find C

SinC/c = sinA/a

SinC = csinA/a = 54.9sin(23.5)/29.2

C = 41.4 degrees or 180-41.4 =138.6 degrees

B=180-A-B= 180-41.4-23.5= 113.1 degrees

or 180-138.6-23.5 = 180-162.1 = 17.9 degrees

b/sinB = a/sinA

b =sin17.9(29.2)/sin23.5 = 22.4 = small b

b=sin113.1(29.2)/sin23.5 = 67.1 = large b

sinC/54.9 = sin23.5/29.2

sinC = .75

C = 54.9(sin23.5)/29.2 =41.4 degrees = small C

C=180-41.4 = 138.6 = large C