Day 77: Generalizing in Trigonometry
Today, in precalculus, we began our study of trigonometric equations…after we shared about an exciting thing we did during break, of course. In the past, I started by giving the students the basic “rules” for finding the principle solutions for the sine, cosine and tangent functions followed by the “special cases” and supplemented with diagrams of the unit circle and parent function. In other words, lots of abstract “stuff” that they should know but quite dry and “stand-and-deliverish.” I was so proud of the graphic organizer I had developed. But every year, I am always surprised by the difficulty some students have with simply applying a standard and rarely deviating process.
So I decided to try something a little bit different. Instead, I started with the problem: cosx = -√2/2. And I asked, what is/are the solution(s) for this equation? Students could give me the first solution right away: 3π/4. And soon after, the value 5π/4. They then said there were an infinite number of solutions because of coterminal angles. So we set about trying to describe all of these solutions using a rule or formula. With little effort, they came up with x = 3π/4 + 2πn and x = 5π/4 + 2πn. Yay! We then connected these values to the unit circle, and then the graph of the parent function.
I then asked, “what are the solutions to cosx = -5/7”? This drew out the inverse cosine idea and the need for a calculator. And they easily arrived at x = 2.3664 + 2πn. But they were unsure how to get the second set of solutions. So we looked at the first problem and discussed how they determined the 5π/4 value and could we use the result 3π/4(from applying inverse cosine) to get this value 5π/4? They tried some things but when we changed the location of the point on the unit circle, their conjectures didn’t work consistently. So we they went back to the unit circle and the solution 3π/4. Eventually they realized that 5π/4 was coterminal with -3π/4….and then we were “off to the races.” Would this work for any point? How could we use this to find the second solution rule for cosx = -5/7? Can we generalize this idea for every “cosx = #” type problem? One thing I pushed them to realize that the -cos-1(#) +2πn comes from subtracting the original solution from zero: 0 – cos-1(#) +2πn. This helped with the subsequent discussion for sine equations. We then went through a similar discussion for “sinx = #” and “tanx = #” and developed generalizations for each.
Finally, I asked how can we show this algebraically? And they went through the process without a hitch. We were even able to talk about the general solutions and those particular solutions required on a given interval. It was nice to determine the solutions and then use the hand-drawn graph to verify solutions with them. I’m looking forward to their experiences with the homework problems tomorrow.
I wonder if this conceptual, student developed generalizations will work better than the graphic organizer. Although this student-centered approach resonates with me, I worry that the time spent works for some but not all students; others also need the graphic organizer….maybe I’ll hand it out toward the end of this topic and have students fill it out on their own. What is your take on using student dialogue and sense-making in your classroom?