Question 1093693: Solve using the Law of Sines and a scaled drawing. If two triangles exist, solve both completely. Round to the nearest tenth.
side a = 23.9 mi
∠B = 64°
side b = 22.0 mi
Select one:
a. ∠A = 77.5°, ∠C = 38.5°, c = 15.2 mi
b. ∠A = 102.5°, ∠C = 13.5°, c = 5.7 mi
c. ∠A = 77.5°, ∠C = 38.5°, c = 15.2 mi or ∠A = 102.5°, ∠C = 13.5°, c = 5.7 m
d. Not possible
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! law of sines says:
a / sin(A) = b / sin(B) = c / sin(C)
you are given:
B = 64 degrees
b = 22
a = 23.9
a / sin(A) = b / sin(B) becomes 23.9 / sin(A) = 22 / sin(64)
solve for sin(A) to get sin(A) = 23.9 * sin(64) / 22 = .9764
solve for A to get A = arcsin(.9764) = 77.5 degrees.
C is equal to 180 - 77.5 - 64 = 38.5 degrees.
c / sin(C) = b / sin(B) becomes c / sin(38.5) = 22 / sin(64)
solve for c to get c = 22 * sin(38.5) / sin(64) = 15.2
one solution is:
a. ∠A = 77.5°, ∠C = 38.5°, c = 15.2 mi
since the given angle B is not included between the given sides a and b, then there is the possibility of another solution.
that solution could be in the second quadrant.
check for a solution in the second quadrant as follows:
180 - 77.5 = 102.5 in the second quadrant.
that means that A is possibly 102.5
if possible, C would be equal to 180 - 102.5 - 64 = 13.5
that's possible so there is a second possible solution.
if you tried to do the same thing with 38.5, then you would have found it is not possible.
180 - 38.5 = 141.5 + 64 = something greater than 180, therefore not possible.
solve for c as follows:
c / sin(C) = b / sin(B) which becomes c/sin(13.5) = 22/sin(64).
solve for c to get c = 22 * sin(13.5) / sin(64) = 5.7
your second possible solution is:
b. ∠A = 102.5°, ∠C = 13.5°, c = 5.7 mi
since both a. and b. are possible, and they are both included in c., then your solution is:
c. ∠A = 77.5°, ∠C = 38.5°, c = 15.2 mi or ∠A = 102.5°, ∠C = 13.5°, c = 5.7 m
a reasonably scaled diagram of your solutions is shown below:
in this diagram:
angle A is on top.
Angle B is lower left.
Angle C is lower right.
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