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# Solving Basic Trigonometric Equations

Lesson 8 of 16

## Objective: SWBAT use inverse operations to solve trigonometric equations.

## Big Idea: Students solve basic trig equations and face a great challenge in finding ALL the solutions by playing Around the World.

*55 minutes*

#### Warm-Up

*10 min*

To warm-up for today’s lesson, I want students to struggle for a bit with the fact that we can have infinite solutions to a trigonometric equation if we have no restrictions on the problem. To do this I am going to have students play an around the world game. Directions for this game are on page 2 of today’s flipchart, Solving Basic Trigonometric Equations. The problem is on page 3.

Check out this video for more details: Solving Basic Trigonometric Equations Around the World Warm-up

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#### Explanation

*40 min*

During today’s lesson, I will lead students through Notes: Solving Basic Trig Equations. These notes were created by www.emathinstruction.com. The notes outline solving trigonometric equations in a way that works for my students. They also offer homework assignments that go along with the notes.

**Special Note**: www.emathinstruction.com posts their instructional resources for free, but their answer keys need to be purchased. Visit their site to learn more about purchasing answer keys.

Today's notes lead students through solving trigonometric equations by understanding the relationship between any given angle and its reference angle (MP7). It is very important that students understand that the trigonometric ratios for these two angles will always have the same absolute value. This is illustrated for students in Exercise 1. In class, I plan to model Problem (a), then I will ask my students to complete parts (b) and (c).

I also plan to model Exercise 2c for my students. I will draw two different rotation diagrams, on two different coordinate planes for the angles. My students had difficulty finding the measure of angle theta after they drew their rotation diagrams, and, wrote in the reference angle. To me, this seems so obvious:

- If the angle is in quadrant I, it will be the reference angle.
- If it’s in quadrant II, it will be 180 degrees minus the reference angle.
- If it’s in quadrant III, it would be 180 degrees plus the reference angle.
- And if it’s in the fourth quadrant it is 360 degrees minus the reference angle.

I don't intend for my students to memorize the above. I want them to be able to recognize these relationships from the rotation diagram (MP5). Sometimes, simple hints can make a big difference. My first move is to remind students that each quadrant is 90 degrees. If this doesn’t help, I ask students to "count it" with me from the initial side of the angle to the terminal side.

One common error in my classroom is students want to add the reference angle and 180 degrees when the angle falls is in Quadrant II. I find that establishing a consistent variable for the reference angle helps students to avoid this tendency. I found this helped my students to better see that the reference angle isn’t necessarily the measure of the angle. The measure will be determined by the quadrant the angle falls in.

My plan is for my students to work through Exercise 3 individually. I will allow students to seek help from their tablemates if needed. During this time, I will sit down with students who are confused about the concept of reference angles. At this stage, I will help them to draw a rotation diagram. I hope that most really just need to see another example, or want me to verify their work. We will review Exercise 3 before all of the students move on to Exercise 4.

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#### Wrap-Up

*5 min*

As I was not going to have time to go over both questions in Exercise 4 before students left I had them do these as a closure to today’s lesson. I had them do a Think-Pair-Share. I really felt that students should be able to solve the two-step trig equations in Exercise 4 without much guidance. So I first gave students one minute to solve either Exercise 4a or 4b (their choice). Then, I gave students two minutes to partner up with someone solving the same one and compare answers. Before departing, I asked two different pairs of students to present their solutions using the document camera, so that the rest of the class could quickly check their own work.

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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
- UNIT 8: Cyclical Patterns and Periodic Functions
- UNIT 9: Trigonometric Equations
- 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