# Day 106: Hey! You already know this.

I used a “sneaky” opener today to get my students to derive the Pythagorean Identity in precalculus based on what they knew from our study of trigonometry:

Opener:  How are the coordinates of any point on the Unit Circle related to the Pythagorean Theorem?  Include a diagram to show the relationship(s).

Lots of good discussion between the table partners as they came up with the theorem…I think its great when students create the content rather than me tell them, don’t you?  I was so happy that they remembered so much since we hadn’t done much trig (except for the “slid in” occasional problem during other topics).

We used what they remembered and relationships that had been eluded to during our initial study of the Unit Circle and trigonometric functions: odd and even relationships, the cofunction relationships, the reciprocal and quotient relationships.  They were so excited about how “everything was related.”  In particular, the idea behind cofunctions and the names of the cofunctions: sine and cosine, secant and cosecant, tangent and cotangent.  They were pumped!

As we derived the other Pythagorean identities, we looked for patterns in their form and discussed why “memorizing” them wasn’t the best approach but rather deriving and seeing the patterns would reap longer-term retention.  Hopefully tomorrow I’ll see how successful this introduction was for my kiddos.

How do you introduce the trig identities?  What strategies do you use to develop students’ long-term retention and flexibility in anticipating, recognizing and using appropriate identity substitutions?