Solve ΔABC. (Round your answers to two decimal places. If there is no solution, enter NO SOLUTION.)

Question

Solve ΔABC. (Round your answers to two decimal places. If there is no solution, enter NO SOLUTION.)

𝛼 = 43.14°,    a = 5.01,    b = 6.11

smaller c:

c =
𝛽 =	°
𝛾 =	°

larger c:

c =
𝛽 =	°
𝛾 =	°

Answer 1

There are actually two ways to approach this one. You can use the Law of Sines or the Law of Cosines to get the two possibilities:

Law of Sines approach:

We know that a / sinα = b / sinβ, which gives 5.01 / sin(43.14) = 6.11 / sinβ. Cross multiply and use arcsin to solve for β = 56.50. However, since this is the ambiguous case for Law of Sines, you have to also consider that the supplement of 56.50 will have the same sine value. Thus, β can also be 180 - 56.50 = 123.50. From there you can just use 180 - α - β for each of the β-values to get that two possible γ (angle C) values. Once you have each of those, just use Law of Sines one more time to solve for both possible values of side c.

Law of Cosines approach:

If we position the sides and angles in the right spot, knowing we have to use angle α, we get the following:

a² = b² + c² - 2bc·cos(α)

5.01² = 6.11² + c² - 2(6.11)(c)cos(43.14)

Using your calculator to solve the resulting quadratic give the two possible c-values 1.69 or 7.22. Once you have each of those, you can either continue using Law of Cosines to find one of the other missing angles or you can switch over to using Law of Sines like above.

Answer 2

two triangles

missing side and angles:

c=1.693, B=123.497, C= 13.363

for 2 decimals

c= 1.69, B= 123.50, C= 13.36

c= 7.223, B=56.506, C = 80.352

c= 7.22, B- 56.51, C = 80.35

B=Beta, C= Gamma

a= side opposite angle A

b= side opposite angle B

c= side opposite angle C