# Day 110: The Perfect Storm

There are just times in teaching Precalculus (or any subject for that matter) when you get to use the beauty and structure of mathematics to actually reason and create an argument to justify a relationship.  And when you can connect geometry ideas with algebra and trigonometry, it’s like the perfect storm of mathematics!  Aren’t I right?!  Well that’s what my students and I did today.  We derived the sum and difference identities for cosine.  And you know what?!  They were with me AND they were asking pertinent questions.

The opener question was pretty straight forward, but I had a few students share how they proved the identity so they could review before the check-in.  Interestingly, one student treated the identity like an equation, and many students recognized the error.  They were really sweet to the presenting student when pointing out the error and the whole class benefited from the reminder of the difference between an identity and an equation (as per our class exploration/discussion and exit question on Tuesday; see student note).

After completing the opener and finishing the 4-problem check-in on simplifying trig expressions, there were only 20 minutes left when we started with the derivation of cos(u-v).  We began with color coding the diagram to be sure we understood how various parts of the expression were related.

The we plowed through the algebra, with constant reminders of what we were doing and what the various parts of the solution represented.

Then we moved on to cos(u+v).  I let them ponder some, and one student suggested re-writing it as cos(u-(-v))…and that was all it took to derive this one.  I asked the students to work on deriving sin(u+v) and sin(u-v).  Of course, I asked them NOT to go to the internet to see how someone else did it, but to use their own noggin.  They were chuckling out the door.