A. Use the law of cosines:
29² = 34² + c ² - 2•34•c cos(30.3°)
Use a calculator to solve for c ; notice this equation is quadratic, you get two possible solutions, c ≈ 5.973 ≈ 6 or c ≈ 52.738 ≈ 53.
Then use the law of sines to solve for the two possibles measures of angle C :
sin(30.3°)/29 = sin(C )/c
sin(C ) = c sin(30.3°)/29
sin(C ) ≈ 0.1039 or sin(C ) ≈ 0.9175
C ≈ 6.0° or C ≈ 66.6°
The interior angles of any triangle sum to 180°, so the remaining angle has measure either
B = 180° - 30.3° - C
B ≈ 143.7° or B ≈ 83.1°
So to recap,
B ≈ 143.7°, C ≈ 6.0°, c ≈ 6
and
B ≈ 83.1°, C ≈ 66.6°, c ≈ 53
