Does cos(A - B) = cos(A) - cos(B)?
Lesson 4 of 15
Objective: SWBAT use the sum and difference formulas for sine, cosine, and tangent.
A common student misconception is that sin(A + B) = sin A + sin B and that the trig function can be “distributed” to the angle measures inside of the parentheses. I am going to start today’s lesson with that misconception and see if students can come to terms with it.
Students will be given the task worksheet and will be asked to investigate whether the statement cos (A – B) = cos A – cos B is true or false. I make sure to give them enough time to think about a few different ways to know that the relationship cos (A – B) = cos A – cos B is false. Students may find a counterexample to show that it is false, but I will press them to think of this at a more conceptual level. I'll ask them to consider the Ferris wheel problem we studied earlier in the year to give them a different perspective on this relationship.
To build momentum from the Launch, I'll ask a few students to present their findings. I might start with a student who found a specific counterexample and have them show that the relationship does not work. Then, I'll choose another student who thought about the relationship in a more general sense and have them explain their thinking. Students may talk about how the cosine value is the x-value of the angle and that it is not proportional to the angle measure. Students may also bring up the Ferris wheel and how the horizontal distance for every 20° interval is not constant.
Now that students understand that cos(A – B) is not equal to cos A – cos B, we can work on finding the actual value. This is not an easy proof, so I am going to go through it with students. It is probably not a conclusion that most students could reach without a significant amount of scaffolding; however, it is worthwhile to prove these identities because I believe that it helps them retain this knowledge. Also, the formula has meaning to them and is not something that is simply assigned to be memorized.
To get things started, I draw this diagram and talk the class through how I am going to set up the problem. By this point, my students should be able to find the ordered pairs for all of the points. I'll add points P and Q to the diagram first and then talk about how one could rotate arc length PQ clockwise until Q is at the point (1, 0). Then SR is the same length as PQ, so the distance between the points is the same. More on this in the video below.
Once the class realizes that PQ = RS, then I'll ask them how to find those distances. After we set up the distance formula for both, it really becomes just an exercise in algebraic manipulation – the only identity they need to know is that sin2 x + cos2 x = 1. At this point I may choose to let them loose and see if they can simplify the problem fully.
With the derivation of the new formula completed, I will ask the class to find the cosine of 15° to demonstrate how the formula works.
After deriving the cos (A – B) formula, I will provide the rest of the sum and difference formulas for sine, cosine, and tangent. I don’t think it’s imperative to derive the other five formulas; students will have a general gist of where they come from. I might want to derive the cos(A + B) formula using the even and odd function identities, or I might delegate that to a homework problem. For the tan(A + B) formula, I will explain that you could use sin(A + B)/cos(A + B) and that it will simplify to the form they will see in textbooks.
For practice, I will have students find all six trig values for 7pi/12 and all six trig values for 255 degrees.