Solving trigonometric equations with graphs
Lesson 2 of 6
Objective: SWBAT use graphs to identify the solutions to trigonometric equations.
Today's lesson helps students develop an understanding of how to use the graphs of trigonometric equations to determine the number of solutions, and, to identify the solutions of a trigonometric equation over a specified interval. In approaching this problem solving process, I will try to help my students to connect their work today with ideas. They know a lot of techniques that they can use, but the equations in this lesson are unfamiliar.
I will begin by asking students to sketch the graph of y=-2sin(.5x)+1 over a single period of the function. This task establishes the ideas of graphing a trig function and focusing on a particular interval. I give student a couple minutes to graph the function. Then, I will ask a student to share his/her graph. It is important to establish the idea that graphing trigonometric knowledge is shared knowledge within the class, rather than something new that I am going to teach the class. For this reason, if there are questions after the graph is presented I will try to have other students answer the questions. Some common questions are:
- Why is the graph reflected?
- How is the period found?
- When do you shift the graph left or right?
I want my students to realize that their peers can help them, instead of depending on me or the text. I also value students ways of communicating mathematical ideas. Sometimes, a student's explanation will be "stickier" than mine. As students share, however, I listen carefully and I re-voice parts that are incomplete or may lead others towards a misconception.
We worked with a sketch in the Bell Work. As we begin moving forward, I will display a precise graph of the Bell Work problem on the board and ask students the following questions:
- Over the single period graphed, how many times will the value of the function y = -2sin(.5x)+1 equal 1/2? 2? -3? etc. How did you determine your answer?
- For which values of x will the function have a value -1? How you determined your answer?
I plan to ask each question, then let the students think and discuss the answers with each other. After about a minute or so I will ask a student to share an answer and then ask another student if he agrees or disagrees. As needed I will probe for justifications for the answers that my students give.
As students discuss the first question, I am listening to hear someone say, "The graph is higher than 1/2." or "If you draw a line at y=1/2 you can see where the graph is at 1/2." This type of thinking is important for the whole class to hear. I think it is more effective when my students hear these ideas from one of their peers.
When I ask where the graph has a value of -1 some students will say, "At its minimum point," perhaps referencing our work with polynomial equations. This answer provides an opening for us to make progress with writing a general statement for when the function evaluates as -1. I will ask, "How would you describe the minimum point or points to someone who cannot see the graph?" Then, we can discuss the existence of multiple solutions, and, how in this case the minimum was only visible once over a single period.
After we have raised these ideas, I will give students the Introduction to Solving Trigonometric Equations worksheet. For this activity, students work in small groups. As they work I will move around and ask questions like:
- How did you determine the exact value for the solution?
- How could you find the solution without graphing? (this is an extension question for students that are moving quickly through the work)
With 10 minutes left in the period I will bring the class back together. I want to get a collective sense of how my students are thinking about the graphs of these functions. As a group we will look at and discuss Question_4. I will ask several students to share what they put on their paper (I have picked these students as I moved around the room).
As the selected students explain their ideas, I am scanning the room to see how many of the students are following their explanation with understanding. The number of nodding heads will give me a good indication of where we stand as a class.
Next, I will challenge the class to think outside of today's practice. I'll ask:
If we only had the key features of a graph such as the amplitude, period, and shifts could we estimate the number of solutions?
I expect that some students will say if you add the amplitude to the vertical shift and the equation is equal to more than that value than there is no solution. Some may comment that the size of the period will make a difference on how many solutions.
Finally, we'll summarize how to use a graph to find the solution to a trig equation:
How did you use the graph to find the solution? Can this be difficult?
Some students have trouble determining the exact answer because they are not graphing with carefully. Others will say using a calculator would make it easier. I ask if the calculator will give an answer like pi/4?
Our Exit Slip today previews tomorrow's lesson. I ask students to quickly try to solve 2sin x+1=0 without graphing. I have students put their answers on sticky notes. We will look at students' ideas tomorrow.