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Law of Cosines and Pythagorean Theorem

Date: 09/23/97 at 00:15:25
From: m.moll
Subject: Angles in a triangle

I would like to know how to calculate the angles in a triangle if all 
you know is the length of the three sides, the reason being that in a 
rocking boat it is very difficult to measure an angle of drift. 
Additionally, it would be helpful to know how to do this when it's not 
a right triangle.


Date: 09/23/97 at 00:56:40
From: Doctor Pete
Subject: Re: Angles in a triangle

The answer to your question involves the use of trigonometry, in 
particular, the Law of Cosines, which states the following:

In any triangle ABC with sides of length a, b, and c, and 
angles A, B, C,  the relation c^2 = a^2 + b^2 - 2ab Cos[C] is true.

It should be noted that angle "C" is the angle that subtends the side 
of length c, and similarly for angles "A" and "B". So angle ABC = "B" 
is opposite the side of length b.  Therefore, to find an angle given 
three sides, we substitute the values of a, b, and c in the above 
equation and solve for Cos[C].  Generally speaking, we get

              b^2 + c^2 - a^2
     Cos[A] = --------------- ,

              a^2 + c^2 - b^2
     Cos[B] = --------------- ,

              a^2 + b^2 - c^2
     Cos[C] = --------------- .

Now, we take the inverse cosine to obtain the angles A, B, and C.  
Note that if the triangle is right, then say angle C = 90 degrees = 
pi/2 radians.  Then since Cos[90 deg] = 0, it follows that c^2 = 
a^2 + b^2, which is the Pythagorean Theorem.  So the Law of Cosines is 
a generalization of the Pythagorean Theorem.

Furthermore, you will observe that the range of the function f(x) = 
Cos[x] is between -1 and 1, inclusive.  So if the fractions in the 
above expressions are too large or too small, then the inverse cosine 
does not exist, and therefore no such triangle exists. For example, 
say a = 1, b = 2, c = 5. Clearly no such triangle may exist, for the 
length of c exceeds the sum of the other two lengths. And the Law of 
Cosines agrees with this, because 

  Cos[C] = (1^2 + 2^2 - 5^2)/((2)(1)(2)) = -20/4 = -5.  

Since the cosine of any real number can never be -5, angle C does not 

So for an example, say we have a triangle of lengths a = 15, b = 10, 
c = 8, and we wish to find the angles.  The Law of Cosines gives us 
the following values: 

     Cos[A] = -61/160 = -0.38125,
     Cos[B] = 63/80 = 0.7875,
     Cos[C] = 87/100 = 0.87.

Taking the inverse cosine on a calculator, we find (approximately)

     A = 1.96199443701 radians = 112.411132046 degrees
     B = 0.664054277243 radians = 38.0475074536 degrees
     C = 0.515594006246 radians = 29.5413605001 degrees.

And we find that the sum A+B+C = pi/2 radians = 180 degrees.

The Law of Cosines is a bit involved and uses some coordinate 
geometry, but understandable to someone familiar with trigonometry.

-Doctor Pete,  The Math Forum
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Associated Topics:
High School Trigonometry

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