The Uniquess of the Law of Cosines

[e2m2a]

Is the law of cosines the only equation that expresses the relations between the sides and angles of a triangle? (Other than the law of sines). For example: Suppose we have a triangle and sides a,b and the angle opposite of c is known. Then invoking the law of cosines, we know that c sq = a sq + b sq - 2abcos(angle). Is this the only valid expression that exists that tells us the value of c sq? Is it possible that other expressions, for example, (just making this up) that c sq = b * pi - cos(4 * pi) + a sq also gives us the same number determined by the law of cosines? Or, is the law of cosines the only valid expression giving us the correct answer. Has it been proven rigorously in mathematics or can it be proven that this the only unique expression that exists that gives us the right answer? And if this is the case, if an alternative expression is found that gives us the same value of c sq for some given values of a,b and angle, can this only occur if the alternative expression can be algebraically derived from the law of cosines?

 

[PeroK]

Is the law of cosines the only equation that expresses the relations between the sides and angles of a triangle? (Other than the law of sines). For example: Suppose we have a triangle and sides a,b and the angle opposite of c is known. Then invoking the law of cosines, we know that c sq = a sq + b sq - 2abcos(angle). Is this the only valid expression that exists that tells us the value of c sq? Is it possible that other expressions, for example, (just making this up) that c sq = b * pi - cos(4 * pi) + a sq also gives us the same number determined by the law of cosines? Or, is the law of cosines the only valid expression giving us the correct answer. Has it been proven rigorously in mathematics or can it be proven that this the only unique expression that exists that gives us the right answer? And if this is the case, if an alternative expression is found that gives us the same value of c sq for some given values of a,b and angle, can this only occur if the alternative expression can be algebraically derived from the law of cosines?

If you know $a, b$ and $\theta$ then there is a unique value for $c$. You can look at $c$ as a function of three independent variables:

$c = f(a, b, \theta) = a^2 + b^2 - 2ab \cos \theta$

Any other expression that gave you $c$ would have to be numerically equivalent to this one for every value of the variables $a, b, \theta$. In other words, $c$ as a function of these three variables is unique.

You could, of course, find an alternative way to write this expression, but it would simply be an identity based on a variation of $\cos \theta$.

 

[lurflurf]

we have the
Law of cosines
Law of sines
Law of tangents
Law of cotangents
Mollweide's formula
sum of angles is pi

these all give the same information and we could find an infinite number of others though none have made it into the trigonometry books with regularity
I am a fan of the alternate form of the law of cosines
a=b cos C+c cos B
b=a cos C+c cos A
c=a cos B+b cos A
it has the advantage of not having squares but the disadvantage of having five variables instead of four

 

[e2m2a]

interesting. never knew this. thanks