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April 10th, 2017, 09:56 PM  #1 
Newbie Joined: Apr 2017 From: Canada Posts: 22 Thanks: 1  Law of cosines superior formula
I have attached a manipulation of the law of cosines which allows to solve accurately and quickly for not only the side opposite to given angle, but also the adjacent sides. It is a microsoft word file compressed in a zip. Regards, Anton 
April 11th, 2017, 12:13 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 17,722 Thanks: 1359 
You are saying that as a² = b² + c²  2bc cos(A) is a quadratic equation in b, the quadratic formula gives possible values of b in terms of A, a and c. I suggest instead using a/sin(A) = c/sin(C) to find the angle C, then using b = c cos(A) + a cos(C). If A is acute, there may be two possible values for C, and hence two possible values for b. 
April 11th, 2017, 09:55 AM  #3 
Senior Member Joined: Jun 2015 From: England Posts: 642 Thanks: 184 
It is good to see that you have done some thinking on your own, especially in this day and age of 'get an app'. But can I please make a plea for you to post your text directly, not as a doubly encoded zip file of a word file. More folks are likely to read it that way. I don't know if you have heard of either the half angle formula or Napiers tangent rule, but these are alternatives to the sine and cosine rules and used to be used for more efficient calculation in the days before 'ask Siri to do it for you'. The sine rule is, of course, ambiguous in the case of 'given two sides and one not included angle'. The cosine rule is never ambiguous. Ancient wisdom on the solution of triangles decrees as follows (1) Given two (three) angles and one side use the sine rule (2) Given two sides and the included angle use the cosine rule or Napiers tangent rule (3) Given three sides use the cosine rule or the half angle formula. (4) Given two sides and the not included angle use cosine rule. I can expand on these if you are interested. 
April 11th, 2017, 10:28 AM  #4 
Senior Member Joined: Sep 2015 From: CA Posts: 1,299 Thanks: 664  
April 11th, 2017, 12:21 PM  #5 
Newbie Joined: Apr 2017 From: Canada Posts: 22 Thanks: 1 
Thank you very much to everyone that replied. I will post the files in the page from now on. I apologize for not knowing the methods that were mentioned. I'm 14, and selftaught in all the math that I know. I have looked at Napiers halfangle formula. Thank you to everyone that responded. I hope to one day be a mathematician myself. Thank you studiot for your kind response 