Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I've been computing the angles of a triangle with sides a = 17, b = 6 and c = 15 using the law of cosines to find the first angle and then the law of sines to find the other 2. I follow the convention of naming the angles opposite these sides A, B and C respectively. Here are my results:

$ C = \arccos( \frac {6^2+17^2-15^2}{2(6)(17)}) = 60.647$ degrees to 3 d.p.

$ B = \arcsin( \frac {6 \sin C}{15}) = 20.405$ degrees to 3 d.p.

$ A = \arcsin( \frac {17 \sin B}{6}) = 81.051$ degrees to 3 d.p.

Clearly, adding these should give $180$ degrees, but it gives 162 degrees to 3 s.f. Assuming I haven't made any mistakes, the error seems quite high and I'm just wondering if anyone knows why this is? It seems high enough to challenge the validity of the laws.

share|cite|improve this question
Maybe it's because of the ambiguous case, that arises when using the Law of sines. Why don't you do the Law of Cosines two times and subtract from 180° to find the third angle? No need for Law of Sines here. – imranfat Oct 15 '13 at 19:07
The laws are acurate and so is your calculator for the trig terms, but the ambiguous case is the issue – imranfat Oct 15 '13 at 19:16
Note that the error is about $18$ degrees, which is just the difference between $81$ degrees and $180-81=99$ degrees. Those angles have the same sine. You are picking the wrong one. – Ross Millikan Oct 15 '13 at 19:19
Hey Ross, congrats BTW on your repuation points. In essence, isn't picking the "wrong" one inherently related to the ambiguous case due to sinx = sin(180-x) ? – imranfat Oct 15 '13 at 19:21
@imranfat: The arcsin function is defined (so as to be single valued) to return values between $-90$ and $+90$ degrees. The error was made in going from $\sin A= ()$ to $A=\arcsin ()$. Those are not equivalent. It is the same as going from $x^2=2$ to $x=\sqrt 2$ and missing the $\pm$ sign. The calculator is useful and returned the correct answer to the question it was asked. – Ross Millikan Oct 15 '13 at 19:27
up vote 3 down vote accepted

OK, I did the Law of Cosines 3 times and came up with 60.647 , 20.404 and 98.949 respectively for angles A, B and C. Remember, the Law of Cosines does not have an ambiguous case, unlike the Law of Sines. I suspect (without further investigating) that his may be the culprit. My advice: Always use the Law of Cosines whenever you can. In this case, when all sides are known, clearly a case for Law of Cosines

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.