Sum and Difference Identities

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SWBAT apply the sum and difference identities to find the exact values of trigonometric ratios of angles that are not derived from special triangles.

Big Idea

The love story of Sinbad and Cosette helps students to remember and apply the sum and difference identities.


5 minutes

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. 


40 minutes

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 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.

Teacher Note:

I am not posting answer keys to these resource. posts their curriculum for free, but their answer keys need to be purchased. See this link to purchase full answer keys: (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."

Closure + Homework

10 minutes