Question 1105678: Describe how the Law of Cosines can be used to solve the ambiguous case of the oblique triangle ABC, where a = 12 feet, b = 30 feet, and A = 20°. Is the result the same as when the Law of Sines is used to solve the triangle? Describe the advantages and the disadvantages of each method.
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Describe how the Law of Cosines can be used to solve the ambiguous case of the oblique triangle ABC, where a = 12 feet, b = 30 feet, and A = 20°. Is the result the same as when the Law of Sines is used to solve the triangle? Describe the advantages and the disadvantages of each method.
Using Law of Sines::
sin(B)/b = sin(A)/a
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sin(B) = 30[sin(20)/12] = 0.86
B = arcsin(0.86) = 58.77 degrees
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Then C = 180 - (20+58.77) = 101.23
Then c = sin(C)[12/sin(20)] = 34.41
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Using Law of Cosines::
c^2 = a^2 + b^2-2ab*cos(C)
But we don't know "c" or "C"
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So we have to use::
12^2 = 30^2 + c^2 - 2*30c*cos(20)
144 = 900 + c^2 - 60c*0.9397
This is a quadratic in "c" and will have two solutions.
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You would then have to find the two values of "c".
Then find the two angles.
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Looks like first using Law of Sines would be more direct.
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Cheers,
Stan H.
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