Law of Cosines Day 2 of 2
Lesson 9 of 13
Objective: SWBAT use the Law of Cosines to solve problems.
Today's class starts with a problem that reviews what the students discovered yesterday, the Law of Cosines. After about 2 to 3 minutes I have a student write the Law of Cosines on the board. I ask a student to do this to help students who may be struggling. As I move around the room, questions I ask to help students include:
- Have you labeled the diagram to show what you know?
- Does the formula look like what you have? (If students only have one formula in mind and it does not include angle A I refer the students to pg 437 in Larson Precalculus with Limits, 2nd ed.
- What can values can you substitute into the Law of Cosines formula?
Once I know that most students have found a solution, I ask a student to demonstrate his/her proces for solving for Side a. Once the work is shared students may ask any questions about the process. I want students to focus on several parts of the process. If students do not ask followup questions, then I'll ask the class questions like:
- Notice how he/she used order of operations to solve, could he had put the entire right side in the calculator at once?
- Why did he/she use the Law of Sines to find angle B?
- Do you notice how he/she labeled each piece after finding the value? Was that helpful?
- How was the value for angle C determined?
- Should we label the unit of measure for the angles? Does using the appropriate labels help us understand what we have found?
I use this line of questioning to help my students practice and internalize problem solving skills.
I now give students a problem to solve where the 3 sides of a triangle are given. I first ask if the 3 sides will make a triangle. After it is determined that we have a triangle I then ask, "Which angle should we solve find first?"
Often, my students are not sure which angle to find first, so I begin questioning them about how to approach the problem:
- After we find one of the angles, what method will you use to find another angle?
- When we use the Law of Sines, can we have more than one solution?
- If we find the smallest angle first could we have 2 solutions?
- What if we find the largest angle?
By finding the largest angle, even if the angle is acute, we will only have one solution. I plan to discuss why we could not possibly have 2 solutions, even if the largest angle is acute. After discussing why we want to find the largest angle first, I'll have my students work in groups to solve the problem.
As students work, I often see that my students want to subtract 2ab from a^2+b^2. I think that this is an attempt to avoid working with cos C, as a term. When necessary, I encourage students to replace cos C with a variable, such as x, so students can solve without making a mistake. Because this problem challenges my students, the solution is shared with the class on the board. I follow up with questions like:
- How was the 346 found?
- Why didn't he subtract 346 and 330?
- What is meant by cos^(-1)(x)?
- Could he have found angle A next?
I try to have a number of different students answer these questions, so that all students have time to think about the problem solving process we are learning to use with the Law of Cosines.
Many books, including our text, list six formulas for the Law of Cosines. I feel that students only need know one formula, since all the other formulas can be successfully derived by the first formula. To give my students practice deriving the alternate forms, I ask them to work the next problem in groups. Most groups are successful until they get to the second to last step. At this point, I am prepared to have my students look at pg 437 of our text (Larson's Precalculus with Limits 2nd ed.). I want them to work together as a group to assess whether or not their formulas is the same as the form in the book. If not, I encourage them to figure out the last step by studying the formula in the book.
Finally, I ask my students if it is necessary to memorize all 6 formulas. I say, "How can we use the Law of Cosines successfully if we remember only one formula?" Most students will say just switch the parameters around or put you values into the equation and solve for the unknown parameter.
To help my students think about the best method to start a problem, the groups are now given pg. 441 #27-32 from Larson, Precalculus with Limits, 2nd ed. I have the students omit the solving part of the directions. Students will share their answers in a survey. I put a chart on the board to record the results. Each group is given sticky notes. Students are asked to put their names on the sticky notes, determine which method should be used to solve the problem, and record their decision on the sticky note. Then, they should place their sticky notes in the correct box.
After the class has finished with the activity we discuss problems that do not have 100% agreement with respect to the solution process. I make note of the groups who are off track, so that I know who needs some more assistance with the Law of Sines and the Law of Cosines.
Finally, to give my students practice using the Law of Cosines, I plan to give the following problems for students to complete in class or as homework:
Larson, Precalculus with Limits, 2nd ed. p. 441-442 #9, 17, 21, 43 and 49.