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# 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.

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

*55 minutes*

#### Warm-Up

*5 min*

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.

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#### Explanation

*40 min*

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.

**Teacher Note**:

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."

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*Hey Tiffany! Thanks, as always, for sharing such a well-developed and exciting curriculum. I know you have already set up this more direct instruction lesson for sum and difference angle identities, but I was curious if I could do a lesson in a more discovery or conceptual way since sum and difference do come up in calculus, especially with the special cases of double and half angle identities. Anyway, I found this lesson,http://www.sineofthetimes.org/discovering-the-angle-sum-and-difference-identities/, and thought you might want to check it out too. I'm going to try it with my students tomorrow. | 6 months ago | Reply*

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- UNIT 1: Basic Functions and Equations
- UNIT 2: Polynomial Functions and Equations
- UNIT 3: Rational Functions and Equations
- UNIT 4: Exponential Functions and Equations
- UNIT 5: Logarithmic Functions and Equations
- UNIT 6: Conic Sections
- UNIT 7: Rotations and Cyclical Functions
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- UNIT 10: Matrices
- UNIT 11: Review
- UNIT 12: Fundamentals of Trigonometry

- LESSON 1: Discovering Trig Identities (Day 1 of 4)
- LESSON 2: Discovering Trig Identities (Day 2 of 4)
- LESSON 3: Discovering Trig Identities (Day 3 of 4)
- LESSON 4: Discovering Trig Identities (Day 4 of 4)
- LESSON 5: Simplifying Basic Trig Expressions – Connect the Dots
- LESSON 6: Verifying Trig Identities (Day 1 of 2)
- LESSON 7: Verifying Trig Identities (day 2 of 2)
- LESSON 8: Solving Basic Trigonometric Equations
- LESSON 9: Group Quiz - Solving Trig Equations
- LESSON 10: Solving Quadratic Trig Equations (Day 1 of 2)
- LESSON 11: Solving Quadratic Trig Equations (Day 2 of 2)
- LESSON 12: Puzzle - Solving Variety of Trig Equations
- LESSON 13: Sum and Difference Identities
- LESSON 14: Error Analysis - Sum and Difference Identities
- LESSON 15: 4-way Representation - Trig System of Equations
- LESSON 16: Review for test