Sum and Difference Identities
Lesson 13 of 16
Objective: SWBAT apply the sum and difference identities to find the exact values of trigonometric ratios of angles that are not derived from special triangles.
At the start of today's lesson, I ask students to work on simplifying the expression on page 2 of Sum and Difference Identities Flipchart. I intend for the problem to serve as review. The task requires students to review both the Negative Angle Identities and the Pythagorean Identities. I ask students to attempt the problem individually. I want to see how far students can get without assistance. I plan to ask a similar question on an assessment, so I would like to know who is able to recall their identities (or look in their trig book and apply the identities) so that I can plan for review.
For today’s lesson, I lead students through the Sum and Difference Identities by presenting notes using the document camera. I accessed these Notes: Sum and Difference Identities from www.emathinstruction.com. I like the outline and I love the homework assignments that go along with the notes. The homework assignments spiral back through concepts and present tasks in a way that requires students to think about a concept in multiple ways.
I am not posting answer keys to these resource. emathinstruction.com posts their curriculum for free, but their answer keys need to be purchased. See this link to purchase full answer keys: http://www.emathinstruction.com/id3.html (accessed April 8 2014)
I begin my presentation by asking students to recall the Pythagorean Identity. Although they may not realize it, my students already have the knowledge to solve exercise 1 by drawing triangles within the coordinate plane and determining their ratios using right triangle trigonometry. I remind my students that they know how to do solve the problem in this manner. However I want them to solve this problem differently by using the identities (MP1, MP2, MP7). I give my students about two minutes to work through the task. I expect some will get it, but some will not. Once I feel the students are losing momentum, I plan to show the class how to substitute into the identity and solve for cosine.
Before I start the second exercise, I will tell my students the story of Sinbad and Cosette (pages 3-5 on Flipchart). Typically, my students find the story funny and reference it throughout the rest of class. I think it provides a narrative anchor point for internalizing the trig identities. Hopefully, it will help them to better remember the Sum and Difference Identities.
Now, I will introduce the Sum and Difference Identities, using the Notes I guide students through Exercises #2, #3, and #4. When we reach Exercise 4 parts e and f, my students will be applying the Sum and Difference Formulas to evaluate these, not the Double Angle Formula! I do not cover the Double Angle Identities in my class class. Instead, I have students rewrite the problem as a sum of two angles.
Depending on how the lesson starts, I may present my students with a challenge: see if they can figure out how to do Exercise 2. I expect that some of my students will want to use a calculator. If they pursue this strategy, I will ask them to provide me with EXACT answers (requiring radical form). I expect that several of my students will be able to handle this challenge. If necessary, I may hint that,"It might be good to re-write the equation using special angles."