Solving Trigonometric Equations Day 1 of 2

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SWBAT use algebraic concepts to solve trigonometric equations.

Big Idea

How can I solve a trigonometric equation that has an infinite number of solutions?

Bell Work

10 minutes

I start today where we left off in the last lesson. On the board is the prompt:

Solve 2sinx+1=0 on the interval (0,2pi].

I begin here because some students that were not sure how to solve the problem may now know how to solve. I also have the sticky notes from yesterday's closure up for students to look over. I have the class look over the ideas from all three classes. As students look over the sticky notes I allow notes to be added.

After about two minutes of review I take the sticky notes for the class and we look over the ideas.  As we review we discuss if the idea seems reasonable. Students explain why the idea is reasonable or unreasonable. After going through these I see if there is any commonalities with the reasonable answers. My goal in this activity is to have students see that the trigonometric equation is solved just like an algebra equation.

Some common sticky note responses include:

  • Replace sin x with a variable, then solve
  • Take the inverse of both sides
  • Get sin x alone, then take the inverse

Solving algebraically

10 minutes

After discussing the different ideas my students have, I want them to experiment with the methods that they have volunteered. Before doing so, I will ask, "How many solutions should we expect? How did you decide on 2 solutions?" This question helps my students to recall what we discovered yesterday with respect to the number of solutions. I give my students a little time to think and to respond. My goal is for students to use their estimates to identify solution processes that will work and will not work. My goal is for my students to try, evaluate, and narrow down the number of possible methods. I often find that this process surfaces (and help students to correct) misconceptions, such as taking the inverse of sine as a first step in solving a trig equation.

Eventually, the class will usually agree that to two possible methods are available to them:

  • Use substitution to replace the trigonometric function with a variable, then solve the resulting equation using algebra to determine the input that makes the statement true.
  • Use algebraic techniques to solve the equation for the trigonometric function, then apply the inverse trig function to solve. 

As they consider different alternatives, I take all available opportunities to remind students that we are trying to identify the angles in the interval (input values) that have the correct output value for the given trigonometric function. 

From here, I give my students several problems to solve. I will be satisfied if we work and discuss only 1 or 2 examples, but I am prepared to offer students additional practice as needed. 

Solving when domain is unlimited

10 minutes

Once I feel that a majority of the students can solve trigonometric equations with a restricted domain, we will move on to how to solve when there is no limit on the domain. To begin, we will return to the bell problem: 2sinx+1=0. I tell students that I now want to solve this equation when x can be any real number, not just between 0 and 2pi. I'll ask, "How will changing the domain change the number of possible solutions? How can we write all the answers when there is an infinite number of solutions?"

As a class we'll discuss the problem until many of the students are comfortable with the idea that the solutions are all a multiple of 2pi from the angles that we found during the Bell Work.  Using this knowledge, we will discuss how to write a mathematical description of such a solution.

As a followup, we will look at 4sin^2x-3=0. In this case, we will write our answer as a multiple of pi (n*pi), instead of a multiple of 2pi. Generally, I allow students to use 2npi, since this is a new idea. I want them to realize that others may write the answer differently. They need to think about the meaning of the expression in the answer!


10 minutes

As class ends I will give students a final equation to solve. This problem has an unusual domain. I present students with this task, in order to see how comfortable my students are when asked to work with a restricted domain. I will collect students work on this task as an Exit Slip at the end of the lesson. I expect that this problem will challenge students, while serving as a formative assessment. 

For homework, my students are assigned p. 394 #40, 41, 42, and 43 from Larson's Precalculus with Limits. I may change the directions to limit the domain to be from 0 to 2pi, depending on how the lesson progressed.