Double Angle Identities
Lesson 11 of 14
Objective: SWBAT use the sum and difference identities to find the double angle formulas.
As my students arrive in class, they pick up a copy of the Double Angle worksheet. The first question is already posted on the board. I encourage my students to work on this first problem in groups.
As the students work I listen to the conversations. For groups that are stuck, I ask, "How could you rewrite 2A so that you can use the formulas mentioned in the directions?" After students struggle for a few minutes (The amount of time depends on the student. Some students have more perseverance than others.) I provide further assistance, if necessary. I pick a student that has made some progress to give me a hint on what might work. I ask, "How did you start? Why did you do that?"
To conclude today's bell work, I choose a student to go to the board and demonstrate how to verify the identity. I try to pick a student who has been working quietly thus far. Page 1 on the examples shows how a student demonstrated his approach to the problem.
Other double angle problems
Having begun the class with a demonstration, I now ask the students to continue to work on the identities on the Double Angle worksheet. During the Bell Work, they saw an approach to these problems. I expect my students to appreciate the structure of the problem and work through the remaining examples, beginning with the identity for cos(2A).
In problems 4 and 5 of the worksheet students see how cos (2A) can be written in 3 different forms. Rewriting cos^2 A-sin^2 A in the other 2 forms allows students to use Pythagorean identities. This can sometimes challenge students who do not see the structure of the Pythagorean Identities and how those identities can be rearragned. Students share their reasoning with the class after working a few minutes. (see page 2 of the examples).
On page 3 of the examples the students determine the identity for tan (2A). After working with sin (2A) and cos (2A) the students are seeing how to replace u and v in the sum formula. The students find the tan (2A) without many problems.
Once the students verify all the identities, they put these identities on their reference sheets.
The last page of the examples gives the students a problem to work. The students find the sin(2A) after finding the value we discussed whether the result seems reasonable. I want students to think about where the terminal side of angle (2A) is on the coordinate plane.
It is important for students to analyze the accuracy of answers. If a student determines the angle should be in quadrant III but their answer for sin(2A) is positive then they know they have an error. On standardized test using this kind of strategy will help student eliminate answers which is one of the test taking strategies taught to students.
After doing sin(2A) I discuss how you can choose which formula to use for cos (2A). Students need to recognize that using one formula may be easier depending on the information given in the problem.
As class ends I assign page 413, # 12, 14, 38, 40, 42 from Larson "Precalculus with Limits, 2nd edition" These problems give the students some practice on using the double angle formulas we discussed today.
In the above problems students are given information like the practice problems. Problem 14 gives some student trouble. The problem asks students to find sec (2 theta). I ask the students if they can rewrite sec x as another function. Once the students remember that sec x =1/cos x they will say "So I can find cos (2x) then write the reciprocal."
Questions 38-42 requires students to find sin (2u), cos (2u) and tan (2u). I always have students ask if they can just do sin(2u)/cos(2u) to find tan (2u). If a student sees this connection I will first ask why do you think this will work. Once they have justified why the method till work I let them use that method. Common Core is about problem solving and using structure to solve problems. If a students see a method that is mathematically correct then the students should use that method to solve problem.