Law of sines

[Diamon76]

Hello,

I saw a formula on a website (Physics website about optics) and I cannot figure out how they obtained this equation (You can find in the attachment the variables I use) :

[MATH] \dfrac{a+b-c}{\sin{(\alpha)} + \sin{(\beta)} -\sin{(\gamma)}} = \dfrac{c}{\sin{(\gamma)}}[/MATH]
I guess it is about the law of sines from which I can say that :
[MATH] \dfrac{a}{\sin{(\alpha )}} + \dfrac{b}{ \sin{(\beta)} } - \dfrac{c}{\sin{(\gamma)}}= \dfrac{c}{\sin{(\gamma)}}[/MATH]But I cannot see how to obtain the first equation.

Thank you for your future help.
Best regards,
Diamon

 

[Dr.Peterson]

Both of your equations can be proved to be true for any triangle using the law of sines, which I'll write as

[MATH]\dfrac{a}{\sin(\alpha)} = \dfrac{b}{\sin(\beta)} = \dfrac{c}{\sin(\gamma)} = D[/MATH]​

This implies that

[MATH]a = D\sin(\alpha),\ b = D\sin(\beta),\ c = D\sin(\gamma)[/MATH]​

Putting those into the LHS of your first equation,

[MATH]\dfrac{a+b-c}{\sin{(\alpha)} + \sin{(\beta)} -\sin{(\gamma)}} = \dfrac{D\sin(\alpha)+D\sin(\beta)-D\sin(\gamma)}{\sin{(\alpha)} + \sin{(\beta)} -\sin{(\gamma)}} = \dfrac{D(\sin(\alpha)+\sin(\beta)-\sin(\gamma))}{\sin{(\alpha)} + \sin{(\beta)} -\sin{(\gamma)}} = D = \dfrac{c}{\sin{(\gamma)}}[/MATH]​

But you're saying they presented that equation as if it should be accepted without proof?

 

[Diamon76]

Oh thank you so much. I am quite ashamed that I didn't find it .... Yes on the website they started with this formula without any explanations. Anyway Thank you so much

 

[Dr.Peterson]

If you came to the equation with the thought that you can combine equal fractions any way you like (e.g. adding or subtracting their numerators and denominators) without changing the value, this could be considered "obvious". But I had to do at least part of the work before I realized it was; it's not something I do every day.

 

[Steven G]

This formula reminds me of the Law of Tangents and Mollweid's thereorems.
Watch a video I made up a while ago on their proofs.

I can't seem to post a video from my desktop. Is this possible?