Problem One
Remark
This uses the Cosine Law. Two sides and an included angle leaves you with no choice. The good news is that there is never any ambiguity.
Step One
Find the supplement to the 105 degree angle. The supplement will put you inside the triangle which you need to have happen for the cos law.
180 - 105 = 75
Step Two
Set up the Cos Law, You will be solving for b
b^2 = a^2 + c^2 - 2*a*c*Cos(B)
Givens
a =200
B = 75 from step one.
c = 180
Solve
b^2 = 200^2 + 180^2 - 2*180*200*cos(75)
b^2 = 40000 + 32400 - 72000 * cos(75)
b^2 = 72400 - 72000 * 25882
b^2 = 72400 - 18634.97
b^2 =53756.03
b = sqrt(53765.03)
b = 231.9
Problem Two
This is the Sine Law
b / sin(B) = c/Sin(C)
Find C
C = 180 - B - A
C = 180 -37.81 - 113.17
C = 29.02 The sin law will give an ambiguity if you use 12 and 113.17. Better to use an acute angle if you can with the sine law.
Solve
b / sin(B) = c/sin(c)
b / sin(37.81) = 6.33/sin(29.02)
b / 0.6130 = 6.33 / 0.4851
b / 0.6130 = 13.048
b = 13.048 * 0.6130
b = 7.9987 which rounds to 8.
Answer D <<<<
