sines law proof

The goal is to prove the sine law:


where the variables are defined by the triangle


and where R is the radius of the circumcircleMathworldPlanetmath that encloses our triangle.

Let’s add a couple of lines and define more variables.


So, we now know that


and, therefore, we need to prove




From geometryMathworldPlanetmathPlanetmath, we can see that


So the proof is reduced to proving that


This is easily seen as true after examining the top half of the unit circle. So, putting all of our results together, we get

sinAa = yba
sinAa = sin(π-B)b
sinAa = sinBb (1)

The same logic may be followed to show that each of these fractions is also equal to sinCc.

For the final step of the proof, we must show that


We begin by defining our coordinate systemMathworldPlanetmath. For this, it is convenient to find one side that is not shorter than the others and label it with length b. (The concept of a “longest” side is not well defined in equilateral and some isoceles triangles, but there is always at least one side that is not shorter than the others.) We then define our coordinate system such that the corners of the triangle that mark the ends of side b are at the coordinates (0,0) and (b,0). Our third corner (with sides labelled alphbetically clockwise) is at the point (ccosA,csinA). Let the center of our circumcircle be at (x0,y0). We now have

x02+y02 = R2 (2)
(b-x0)2+y02 = R2 (3)
(ccosA-x0)2+(csinA-y0)2 = R2 (4)

as each corner of our triangle is, by definition of the circumcircle, a distanceMathworldPlanetmath R from the circle’s center.

Combining equations (3) and (2), we find

(b-x0)2+y02 = x02+y02
b2-2bx0 = 0
b2 = x0

Substituting this into equation (2) we find that

y02=R2-b24 (5)

Combining equations (4) and (5) leaves us with

(ccosA-x0)2+(csinA-y0)2 = x02+y02
c2cos2A-2x0ccosA+c2sin2A-2y0csinA = 0
c-2x0cosA-2y0sinA = 0
c-bcosA2sinA = y0
(c-bcosA)24sin2A = R2-b24
(c-bcosA)2+b2sin2A = 4R2sin2A
c2-2bccosA+b2 = 4R2sin2A
a2 = 4R2sin2A
asinA = 2R

where we have applied the cosines law in the second to last step.

Title sines law proofPlanetmathPlanetmath
Canonical name SinesLawProof1
Date of creation 2013-03-22 11:57:41
Last modified on 2013-03-22 11:57:41
Owner drini (3)
Last modified by drini (3)
Numerical id 9
Author drini (3)
Entry type Proof
Classification msc 51-00
Related topic CosinesLaw