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May 13th, 2017, 06:37 AM   #1
Ala
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Question

So I'm studying math for tomorrow's exam and got onto this problem.
The book states that you can use the law of Sines
(sinA/a = SinB/b = SinC/c)
in the (AAS, ASA, SSA) conditions to solve the triangle, and use the law of Cosines
( a^2 = b^2 + c^2 - 2bc cosA , ...etc.)
to solve triangles in the (SSS, SAS) conditions.

Now I have this Triangle △ABC:

∠A,C = unknown, ∠B = 36°, b = unknown, c = 5, a = 7.

The triangle here is from the SAS condition so we have to use the law of Cosines to find the unknown values.
To find the missing side (b):
b^2 = a^2 + c^2 -2ac cosB
b^2 = 7^2 + 5^2 -2(7)(5)cos36°
b ≈ 4.2

and here is where the problem starts, since we have now 3 known sides and 1 known angle, we can use both the law of Sines and the law of Cosines to find the remaining Angles, however, when I use the law of Sines I get a different answer from when I use the law of Cosines.

Here, let's try to find ∠A using the law of Cosines:
a^2 = b^2 + c^2 - 2bc cosA
7^2 = 4.2^2 + 5^2 - 2(4.3)(5)cosA
7^2 -4.2^2 - 5^2 = -2(4.3)(5)cosA
cosA = (7^2 - 4.2^2 - 5^2)/[-2(4.3)(5)]
cosA = -0.1514
∠A = inverse cos(-0.1514)
∠A ≈ 99°

and now let's try to find it using the law of Sines:
SinA / a = SinB / b
SinA / 7 = Sin36 / 4.2
SinA = 7 Sin36 / 4.2
SinA = 0.97964
∠A = inverse sin0.97964
∠A ≈ 78°

The answers were different using each law - where's the fault? Is it my calculations? Is it the law was incorrect? Need quick help please.

Last edited by skipjack; May 13th, 2017 at 08:19 AM.
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May 13th, 2017, 09:09 AM   #2
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If you use 4.167 instead of 4.2, you get 80.9°. As the angle is obtuse, you need to subtract this from 180° to get 99.1°, which is about the same as your earlier value.

It isn't immediately obvious that the angle at A is obtuse, but you could calculate the angle at C instead, etc.
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May 13th, 2017, 01:10 PM   #3
Ala
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how did you know that it was obtuse?
when i calculate the C the answer was 44.8 using both laws, and i was able to find A by subtracting B and C from 180 (180 - 36 - 44.8 = 100) which is around 99.1
but i didn't get how you found that A is obtuse and went for the 99.1 over the 80.9
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