Proof pythagorean theorem in extended law of sines

[bartsnapp]

https://www.yahoo.com/entertainment/2-high-schoolers-theyve-found-145534941.html

Recently, high school students Calcea Johnson and Ne'Kiya Jackson at St. Mary's Academy in New Orleans gave a new proof of the Pythagorean theorem.

They were reading a book from 1940 called "The Pythagorean Proposition" that claimed "there is no trigonometric proofs because they are "based upon the truth of the Pythagorean Theorem."

Our young mathematicians proved this book wrong by providing such a proof. Let's see if we can reconstruct their work from the facts above.

For extra credit:

1. Define sine and cosine

2. Prove the identity (pyth id)

with out appealing to the Pythagorean theorem.

3. Condense our proof of the extended law of sines. Note that we don't use the Pythagorean in its proof.

4. Draw a right triangle. Then use the extended law of sines to prove the Pythagorean theorem.