Question

What are the relationships between an angle (in a triangle) and it's opposite side (in a triangle)? Does an angle determine the length of it's opposite side (in a triangle)? Are the answers to these questions the answer to why trigonometry even works? Is this why it works? Is this why trigonometry even exists? I know, this is a very BIG question and it definitely requires much critical thinking, research, effort, and time in order to answer such a broad question. So, I won't mind if nobody bothers to answer this question; I am a very curious person, and I am very eager to learn the accurate answer to the questions.

Answer 1

Triangle ABC, sides a, b, and c are respectively opposite ∠A, ∠B and ∠C.

Using trigonometry, we have that 

         a / sinA  = b / sinB = c / sinC = 2 R
         a = 2R sinA        b = 2 R sin B       c = 2R sin C
 or,    a / b = sinA / sinB 

where R = radius of circumcircle, the circle on which the vertices of the ΔABC lie.

So in a triangle,  if ∠A is larger than ∠B, then the side a is larger than side b.  if ∠A is smaller than ∠B, then a is smaller than b.

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Other than trigonometry, it is simple to see that the side opposite to a larger angle is bigger in a triangle.

     If ∠A is larger than ∠B, it means the two sides b & c enclosing ∠A are wider apart as we move away from vertex A.  The side a is drawn between these two sides.  So obviously the length of the side a becomes bigger.

  The wider apart sides b and c are, the larger the side a becomes.

It is simple.