##
* *Reflection: Complex Tasks
Using Formulas to Solve Trig Equations - Section 1: Launch and Explore

The biggest struggle in finding the correct answers to these trig equations is always considering the period. One thing I tried really hard to stress was that we always start with the period of the original function we are working with (e.g. if we have a cosine equation we consider the period of the cosine function). After we set up a solution like 2x = 45° + 360°,* then* we divide by 2 to get the period of the transformed function. It seemed easier to talk about it in this algebraic way and then to make the connection to the period of the transformed function.

*Finding the Period of the Solutions*

*Complex Tasks: Finding the Period of the Solutions*

# Using Formulas to Solve Trig Equations

Lesson 10 of 15

## Objective: SWBAT solve trig equations that must be simplified using trig identities.

*50 minutes*

#### Launch and Explore

*15 min*

Yesterday we solved equations that involved trigonometry and we found that there were an infinite number of solutions because the trig functions were periodic. Today we are going to build on that concept by continuing to solve trig equations, but today we are going to have to simplify them using the trig identities. Like yesterday, **the most important points** when solving these equations are staying organized and thinking of *every* angle that is a solution, not just the one our calculator gives us.

I give my students this worksheet and have them work with their table for about 15 minutes to get them thinking about the problems. They may notice that these equations look much more complex than yesterday’s. That is a good observation, so I tell them to** try to simplify** them to make them easier. For equation #1, for example, students may not want to work with the equation csc(4x) = 5/3, so I ask them if they could rewrite it using a more familiar trig function. Or for the third equation, many students recognize that the quantity on the left side of the equation can be simplified using the sin(A + B) formula.

I am not looking for students to get all of the correct answers at this point, but to start thinking about these problems. If some students finish a problem entirely, I will suggest that they check their answer on a graphing calculator.

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#### Share

*20 min*

When it is time to share, I will usually begin with equation #3 from the worksheet since it is the most complicated. I will choose a student who recognized the need to simplify using the **sin(A + B) formula** and have them share their thoughts. After the equation has been simplified to sin(2x + 10°) = sqrt(2)/2, then we can use inverses to find the two angles expressions that would work. Here is an example of how to keep the work organized so no solutions are missed.

One **common mistake** is that students will use the (2x + 10°) = 45° + 360°n equation and solve for x to get x = 17.5° + 180°n. Then, they will use 17.5° to find another angle (162.5°) that has the same y-coordinate, and think that it is other correct answer. If students plug it in to the original equation, they will find that it does not work. I stress that it is important to find the two angle expressions before we algebraically solve for x.

After our algebraic method of solving, I put up this graph and ask students how we could use it to solve the equation. Then I ask them what would happen if we did not set up the second angle expression 135° + 360°n = (2x + 10°). Many students will realize that we would be missing half of the solutions. They may also notice that we would only be getting the solution on the left side of each peak, not on the right side.

There may not be time to go over the other two equations completely, but I at least get them going on the right track by having students explain what their thinking was to simplify. For example, for the second equation, choose a student who substituted 2cos^{2}x – 1 in for cos(2x) and then factored. This will be enough to get students on the right track if they are not already.

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#### Summarize

*15 min*

To end this lesson, I ask students to** reflect on the three equations** and ask what was the important step to simplifying all of these equations so that it could be rewritten as something we knew how to work with. During the discussion, most students realize that we had to use one of trig identities to rewrite it in a simpler form. I stress that these trig identities are tools that can be helpful to us, and allow us to solve equations that we wouldn't be able to solve otherwise.

Here is an assignment for students to get some practice with solving trig equations. I discuss one of the questions in the video below.

#### Resources

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- UNIT 1: Functioning with Functions
- UNIT 2: Polynomial and Rational Functions
- UNIT 3: Exponential and Logarithmic Functions
- UNIT 4: Trigonometric Functions
- UNIT 5: Trigonometric Relationships
- UNIT 6: Additional Trigonometry Topics
- UNIT 7: Midterm Review and Exam
- UNIT 8: Matrices and Systems
- UNIT 9: Sequences and Series
- UNIT 10: Conic Sections
- UNIT 11: Parametric Equations and Polar Coordinates
- UNIT 12: Math in 3D
- UNIT 13: Limits and Derivatives

- LESSON 1: Trigonometric Identities - Day 1 of 2
- LESSON 2: Trigonometric Identities - Day 2 of 2
- LESSON 3: Student Work Day and Individual Conferences
- LESSON 4: Does cos(A - B) = cos(A) - cos(B)?
- LESSON 5: If sin(A) = 3/5, what is sin(2A)?
- LESSON 6: Formative Assessment Review
- LESSON 7: Formative Assessment: Simplifying Identities and Trig Formulas
- LESSON 8: What is cos(22.5°)?
- LESSON 9: Solving Trig Equations
- LESSON 10: Using Formulas to Solve Trig Equations
- LESSON 11: Extraneous Solutions
- LESSON 12: Putting All of the Pieces Together
- LESSON 13: Formative Assessment: Solving Trig Equations
- LESSON 14: Unit Review Game: Lingo
- LESSON 15: Unit Assessment: Trigonometric Relationships