What is cos(22.5°)?
Lesson 8 of 15
Objective: SWBAT derive and utilize the half angle formulas for sine and cosine.
The goal for today is to derive a formula to find cos(1/2A) given cos(A); we will also find the formula for sin(1/2A) and tan(1/2A). Similar to the lesson where we found the double angle formulas, the focus will be on starting with a common misconception and the debunking it and trying to find the correct answer.
To begin the lesson, I give students the notes worksheet and have them work on #1 in their table group. The focus is on MP3 (critique the reasoning of others) as students are going to respond to Gail’s argument that cos(22.5°) is half of cos(45°). I give students about 7 minutes to respond to her reasoning. If they agree I ask them to prove that it is true. If they disagree I ask them to give reasons why it is false.
Most students will find out soon enough that Gail is incorrect. Their reasons for stating that this is false are usually varied, but here are some good ways to make it convincing. I will hand pick students with each one of these arguments.
- The most common way they do this is to just plug both into their calculator and notice that cos(22.5°) is not half of cos(45°). Students may also try this for other angle measures.
- Students think about what cosine represents (the x-coordinate on the unit circle) and reason that when the angle measure is divided in half, the x-coordinate is not half of what it was originally. In some cases (Quadrant II for example), the x-coordinate will actually increase when you take half of the original angle.
- Students may think about the graph of y = cos(x) and y = cos(1/2x). The ½ in the second equation means that the graph is stretched horizontally. The amplitude will not change, so it wouldn’t make sense that the cosine value will be half of the original.
If these reasons do not come out, I will try to lead the class there. Because this is such a common misconception, it is important to refute it in as many ways as possible to make it most convincing to all students.
After this initial investigation about Gail’s reasoning, students are hopefully convinced that cos(22.5°) is not equal to ½ of cos(45°). Now we want to focus on finding the correct answer. I believe that if students can pinpoint which existing formula will help them find cos(22.5°), then they can probably do this on their own. I give them about 10 minutes to work on questions #2 and #3 on the worksheet.
Students are usually drawn to the double angle formula, because they know cos(45°) is double cos(22.5°). The tricky part is the substitution. In the formula cos(2x) = 2cos2(x) – 1, we must replace the (2x) with 45° and the (x) with 22.5°. Once students do this, they can algebraically solve for cos(22.5°). Here is our work from class.
Once they square root both sides of the equation, students will usually wonder about the plus or minus. When students get to this point, it is a good idea to gather the call the class back together and discuss this as a class. I always point out that we choose plus or minus based on what sine the half angle should be, not the original angle. A common misconception is that students will choose the plus or minus based on the original angle.
Share and Summarize
After our initial work with finding cos(22.5°), I won’t derive the formulas for sin(1/2x) and tan(1/2x) in the interest of time. I think one derivation is sufficient for students to get the gist of this process. However, you could always assign it as a homework question. So for question #4 on the worksheet, I will simply give the students the correct formulas.
When we move on to question #5, the discussion for choosing plus or minus will be more pressing. We don’t even know what the measure of angle A is, so I will ask students to several different angles in the given interval and they should notice that half of A will always be in quadrant II. Students will work on this one and I will select students to share their work. Sometimes I will choose a student who chose the wrong plus or minus sign to see if the class can catch their mistake.
In this video, I give some ideas on how to wrap up this lesson.