Sum and Difference Day 1 of 2
Lesson 9 of 14
Objective: SWBAT prove the sum and difference formulas for sine, cosine and tangent
I begin class by giving my students a situation to consider.
In our first unit of the year on Functions, we discussed how f(x+y) does not always equal f(x)+f(y). Some of my students will remember this idea, but many will need to be reminded. With respect to the situation under consideration, I tell my students to use a calculator to explore some ideas using approximations.
After 3 or 4 minutes of work I ask, "What have you tried to do?" I expect that a majority of the students will have tried sin 30 + sin 45. However, I usually find that students do not want to share what did not work. With respect to the development of mathematical practices, however, it is important for the students to test hypotheses and share what they have determined. By finding that an initial hypothesis is false and discussing why, I think it allows my students to remember more than if I just tell the students it will not work.
So, we begin our discussion by sharing what students have tried so far. As we discuss why some ideas do not work, I will offer constructive feedback. I expect that one or more students will produce the correct formula for sin(A+B). When this happens I will ask, "How do you know that is the correct formula? How did you find the formula?" It may be the case that these students found the formula in the book or they have used it in an engineering course at my school. I continue to question by saying "Do you know why it works? Let's explore why it works."
I let this conversation proceed until we reach the point that the students have exhausted all ideas. Then, I am ready to present a a formal argument for sin (A+B).
For this proof I decide to use a Kahn Academy video. I want students to be able to see the proof, and, to be able to review the presentation. I also want students learn how they can use online videos when they are absent or when they are in college. At first, many students will not think about stopping the videoing and analyzing what is being said. They may watch the video once say they are confused. So, before starting the video I discuss how I am going present the video, as if I were watching the video to understand and learn from another person's reasoning (MP3).
Then, we watch the video. I stop the video at several points to give students time to process the argument. We discuss ideas that may be confusing. I sometimes ask students to predict what may happen next to see how well they are understanding the material.
Here is an outline I use to remind me where to stop. It includes questions I will ask students at different points in the video. The outline identifies points where I think students may need assistance. I lead this presentation flexibly. I sometimes do not stop at a refresher point (i.e 1:12 and 3:04). I also write ideas on the board if I sense that students are confused. Every class has different needs so I try to anticipate what will help the students the most and change my presentation to meet the students' needs.
After watching the video I ask each student to put the formula on their reference sheet. I have the students write the identity as sin (u+v)= sin u cos v +cos u sin v so that we can use this identity to prove other identities.
We then use the identity (page 2) to find the exact value of sin (75). As you see on page 2, I have the students identify the value of angle u and angle v. I expect some students to struggle with substituting in the formula. I always write out the formula and then identify the u and v. In the next lesson we will write out what will replace sin u, cos u, etc.
We verify the result by find approximating the formula answer with the calculator answer for sin 75.
Students will now use the sin(u+v) to find the other sum and difference formulas. I give students Finding the other sum and difference formula worksheet.
I give the students a few minutes (3-4) to consider what should be done. If I see that students are failing to make progress, I will bring the class back together. I ask the students, "What does the worksheet say to do? What does that mean?"
We will write out the original identity and then what we are to find. One the other sum and difference resource you can see on page one who the class progresses. Some students will struggle at first with writing A-B as A+(-B). I will use numbers (such as 3-4=3+(-4)) to help them see what we are doing. After substituting in the -B students begin to see the even/odd identities and change the equation.
A few students will write cos A-sin B instead of cos A(-sin B) or -cos A sin B. For some reason when the trigonometric function is in a problem they do not see how this is just like x(-y). Students struggle with connecting the what they call "algebra" (using x and y with the trigonometric functions. I have to show students the situation in terms of single variables to assist with the understanding of the structure of the original problem. In this case I write x(-y) and ask how we can rearrange this to a common notation.
On page 2 of other sum and difference resource, I help the students with the directions in the problem. The students need help in grouping the 3 angles so that they can simplify the expression. Once the grouping is done I make sure the students identify the u and v. Some students will need reminded about the co-function identities.
As we work I have the students identify the u and v and substitute the values into the formula. The students will then use other properties to simplify the expressions.
In question 4 or the worksheet and page 4 of the examples, I give the students the final answer since the will need to use some algebra strategies to find the answer.
I use the method of giving students a few minutes (at least 5) to work on the question. I then have students share their work on the board. The students will struggle determining the final step of the verification.
I demonstrate my method by explaining that I look over what I need and what I have. I notice that I need a 1 in the denominator so I ask myself, "what can I do with cos A cos B to make it 1?" Once it is determined that we divide by cos (A) * cos (B), I will ask, "if we divide that term by cos A cos B what do we do with the remaining expression so we do not change the value?" I give students time to determine that we divide all the pieces. The red text on the work page shows what the students did. After writing this students begin explaining that factors will cancel and we have either tan A or tan B left.
After discussing most of the formulas students are left to find the formula for tan(A-B). Students should be able to use the structure of the prior problems to find this formula. At the end of the lesson, I encourage students to make sure they have all the formulas on their reference sheets.