Proving Math Formulas to High School Students

I am teaching PreCalc this year.  We are knee deep in our trig unit and I am planning to teach Trig Angle Addition formulas on Monday.  I seriously dread this day because I don't see any motivation for it.  I mean we have a calculator to figure out what sin 75˚ or cos 165˚ is...who cares about the exact value???

So, I checked around the internet to see if I could find some motivation and I seriously still haven't found too many reasons why we are still actually teaching this topic, BUT I did find some motivation for me!  I found several different (but similar) proofs of the formulas:

sin(x+y) = sin x cos y + cos x sin y

cos(x + y) = cos x cos y - sin y sin x

Can I tell you a secret?  I never had any idea where these formulas came from.  As a math teacher, I'm embarrassed to say that, but it is true!

So, I decided to give this proof a try during class last Monday.  It went about as I expected - some students appreciated the proof...some students' eyes started to glaze over.  But, I felt like I had done my duty and showed them some of the the beauty of mathematics.  Formulas that we use come from unlikely places - someone had to think of them...they didn't just appear out of thin air.

Here are links to a few products that I have that encourage students to see the "proof" in mathematics.  What proofs do you show your class?

An InLinkz Link-up