Because we’ve already spent time verifying the Pythagorean trig identities, I choose to focus this lesson more on using those and other fundamental trig identities to simplify and find exact values for trigonometric expressions. I begin this lesson with the identities we’ve already discussed on the board along with the co-function and negative angle identities. I also post the question “How are these equations related?” There is a photo of this board entitled “Identities” in my resources for this lesson. To begin class I ask my students to think-pair-share with their left shoulder partner about the information on the board. This allows them to build their MP7skills as they look for similarities and differences between the equations. It also reinforces their general mathematical communication skills as they talk with their partner. While they’re discussing, I walk around and make note of any particularly strong observations as well as general trends in the discussion. As the discussions wind down, I call the class back to a full-class discussion and share what I’ve heard, without naming specific students or teams. For example, I might say “I heard one group talking about how all the equations use trig functions without any values given so I don’t see how we can use them to solve anything” or “Another group asked if any of these equations was over a specific domain for θ.” I then call on students randomly to give their response to the statements, an opportunity to work on MP3 skills. If none of these comments were actually made, I either lead my class to them using guided questioning, or I state them as having been said and move on from there. I optimistically anticipate my students will be able to reach a consensus about the equations and recognize that they are all different ways of organizing the relationships between the trigonometric functions. That leads us to our main activity of the day, working with these identities.
You will need copies of the “Homework” handout for this section. I close this lesson by assigning specific problems from the text that I’ve selected for their relevance and rigor. I am not a fan of simply assigning the evens with the hope that extensive repetition will make the content understandable, so I try to choose problems carefully and always expect my students to show their thinking both mathematically and in writing. The first three problems allow my students an opportunity to practice simplifying expressions using the double and half angle formulas without having to worry about going beyond that step, but they also each incorporate more than just a one-step simplification and reinforce an understanding of the relationships between the trig functions. The next three problems give my students a different perspective on manipulating quantities and also on understanding limits to the domain of the function. The final four problems take my students to the next level by asking them to verify trig formulas and to use those formulas to simplify and solve a real-world problem. Verifying may seem like a make-work assignment, but I appreciate the ways my students find to accomplish this task and the depth of understanding it helps them build for themselves.