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.
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 2cos2x – 1 in for cos(2x) and then factored. This will be enough to get students on the right track if they are not already.
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.