Rise and Sine

The Best Way to Teach Trig Identities to Your Precalc Class

Hi, my name is Tara, and teaching trig identities is my favorite subject! It wasn’t always that way. And truthfully, it probably took a good 3 or 4 years of revisions to develop the trig identities unit that I loved to teach and that my students really succeeded at. Without a textbook, I just had to go for it. At first, I assigned many lessons that were too long for the class period. Or had an answer key with so many errors. Or just assigned identities that were too hard, too soon. I’d mark my failures with a post-it note to myself and revise for the following year. Finally, my all-time favorite unit to teach was born! And because I love to save teachers time, I’m going to share the bones of that unit with you, too!

Start with Fundamental Trig Identities

First, you’ll need to start with the basics. For the first lesson, I start with reciprocal and quotient identities. And the goal is just to simplify each expression to a single term. 

A list of reciprocal identities and quotient identities that make a great first lesson for when you teach trig identities.
Simplifying trig expressions using reciprocal and quotient identities makes the perfect first lesson.

Pythagorean Identities

Since it’s a CCSS, you’ll likely need to incorporate deriving the Pythagorean Identities into your lesson. Depending on the strength of my class I either include it in the notes or sometimes I utilize this interactive notebook page to help isolate each step. Be sure to go one step further and rewrite each Pythagorean Identity for each term. So in total, your students will have nine to reference. 

Derive the three Pythagorean identities and then rewrite for each term. Students will have nine identities to reference.
Derive the three Pythagorean identities then rewrite for each term.

Factoring with Trig Identities

The key to incorporating trig identities that use algebra skills is … review those algebra skills! Each example I show demonstrates with an algebra example, followed by an example that mirrors the step with trig terms. Sometimes (ok most of the time) I even supplement a mini Algebra Review Unit before starting Trig Identities at all. 

Trig Identities Using Complex Fractions

Similar to factoring with identities, each example I present will first be done with just algebra. Students will need to remember the different ways to achieve common denominators. Then I provide a similar example with trig terms.

Here’s a good place to review, review, review! Through this point, the objective is to simplify each trig expression. This is a good spot to pause and have a quiz. 

Review algebra concepts before applying to trig expressions

Time to Verify Identities

Now that your students should be comfortable with simplifying trig expressions in various ways, you can introduce the concept of verifying trig identities. I tell my students to pick the more complicated side of the equation, and pretend it’s just a trig expression to simplify! I shy away from using the term “proving” trig identities because they typically associate geometry proofs with the word prove and lots of groans will ensue. You have been warned – ha!

Double Angle & Sum and Difference Identities

First, we reviewed all the double angle and sum and difference formulas. Then I showed an example of each. In our class, we did not do half-angle formulas, but you could do double angle and half-angle in one day, and then a separate day for sum and difference formulas. 

My students spend a lot of time practicing – a lot. I don’t know how many identities my students verify – throughout the whole unit, it has to be close to 100! But with all this practice students notice similarities from identities they have already completed, and with an organized binder they begin to flip through the notes and find the strategies that apply to that identity. 

Double angle & sum and difference identities make for a good challenge at the end of the unit

How to Assess?

Typically my precalculus classes were juniors and seniors that were not on the route to AP Calc but were still decent math students. Their motivation was typically a bigger struggle than their ability. When I created my unit test, I made the verification portion a choice board. I provided 6 challenging identities to verify, but they only needed to choose 4 of them to complete. Once I implemented this strategy my assessment scores went way up. Plus, most students completed all 6 questions on scrap paper and chose the 4 they were most confident in. Their feedback to me was the assessment was challenging but fair, and that’s exactly what my teacher heart likes to hear. 

I also provide a blank reference sheet for students to use on all assessments. Students are required to fill it in so if this is something you want to do be sure to design an assessment that leaves time for them to use the reference sheet.

You can grab a free reference sheet! Just tell me below where to send it.

Would you like to try my Trig Identities Unit?

Here’s the FULL TRIG IDENTITIES UNIT! Or grab each lesson individually:

I also have some digital resources you can use as extra practice!

I hope this was helpful for your planning! Let me know what your students struggle with. I love having new topics to write about.

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