Intuition For The Law Of Sines

This is a companion discussion topic for the original entry at

Do each possible triangles within the circle have their perpendicular bisectors intersect at the center of the circle?

thanks for another fantastic article . by your analogy the giant gorilla is tringle , angles are kids that see different versions (different sides) of gorilla , the angle of window and gorilla,s length never changes but why a circle circumscribe three windows , all possible triangle ???

hi khalid i studied all of your article i just want to say thanks. i can calculate the amount of mass sucked by a black hole from a donar star yet when i read about your articles about basic stuff like e and gradient i feel like i am ((understanding)) the concepts for the first time. i just want to say thanks for your great site.
since you are excited about understanding basic mathematics, i suggest you to read ((what is mathematics by richard courant ))

This is superb. With the usual proof I feel like someone with a better grasp of the ‘behind the scenes’ maths spotted a trick I wouldn’t find. With this, you can see the process of discovery. Somewhat different to a proof, but much more fit for purpose for the classroom. Thanks!

Thanks Anthony – exactly, I prefer to share the discovery process (“How did we get here?”) vs. the super-concise, cleaned-up proof that might not be very illuminating. Dropping right triangles, re-arranging, etc. doesn’t make sense without that “Aha! There’s a shared circle!”.

This is awesome. Could you use the same analysis/analogy to talk about cosine and tangent also?

Great question. For this scenario, sine is the only ratio that matches up with the fixed side. You could use trig identities like

$$cos^2+ sin^2 = 1$$

and rewrite the Law of Sines as:

$$\frac{a}{\sqrt{1 - \cos^2(A)}} = \frac{b}{\sqrt{1 - cos^2(B)}}$$

but it doesn’t have the right zing :). But there’s likely other geometric connections with the cosine or tangent that can be expressed more simply.

Kalid. This is AWESOME. Will definitely be using it in the classroom! (and thank you for improving MY understanding of the law of Sines : ). O

Hey Ollie! That’s awesome to hear. I’ve only recently figured out the meaning myself, really happy if it’s useful for your class :).

Excellent article … helped a lot in intuitive understanding.

it is absolutely fantastic! You have helped lift up burden of memorization. I am increasingly getting contagious,no longer satisfy with mathematical expressions until I gain insight into the concept and its generalization (abstraction). I don’t have this behaviour before, it started after getting baptized into your site. I have read through the book suggested by Mahmud, “what is mathematics” by Richard Courant. Your site as equally drove me to read “Men of mathematics” by E.T. Bell. Thanks alot for showing the way.

This can be also be used to prove that sin(180-θ) = sinθ