Operations of Complex Numbers in Trigonometric Form
Lesson 7 of 11
Objective: SWBAT multiply and divide complex numbers in trigonometric form.
Yesterday students found the trigonometric form of complex numbers. Today students see how complex numbers in trigonometric form can make multiplying and dividing easier. I start by giving students 2 complex numbers to convert to trigonometric form
To make the work quicker I divide. Each students receives a card with either a #1 or #2 on the card. Students work the problem and compare with someone who has the same number. This cooperative activity helps students learn to communicate with other students in their class. After students have compared with at least one person (3-4 minutes),l I pick 2 students to share the answers.
After sharing the trigonometric form of the equations I continue using the groups developed in the bell work throughout the class. Using the standard complex form of the numbers, Group 1 finds the product of the numbers while group 2 finds the quotient by rationalizing the denominator. Once the product and quotient are found students put the results in trigonometric form. These results are also shared with the class.
Once all the results are on the board, I am ready to have students to determine how to use the trigonometric form to find the product and quotient of complex numbers. I put all the trigonometric forms on the board. I now ask students to compare the 2 original vectors with the product.
- Is there a way we can use the trigonometric forms to find the product?
- Look at the coefficients what do you notice?
- What about the angles what do you notice?
Students see how the coefficient of the product is the product of the complex number coefficient but have a little trouble seeing the angle relationship. I guide the class by asking "Is there another angle that is at the same position as zero degrees? Let's rewrite the product with that angle? Do you see a relationship now?" Once we change the angle to 360 degrees students see how the product angle is the sum of the original angles.
Using what you just noticed could a rule be developed for finding the product if the complex numbers are in trigonometric form? As a class the students determine a rule.
I use the same strategy as above to find a rule for division. Students discuss ideas for 3-4 minutes and then share out ideas. Once we verify the rule we write it out on the board.
It is now time to have students practice using the two rules they have found. Students work on a problem. When I do this problem I give students the product first. After finding the trigonometric form I have student convert to standard form. This is to remind students how this is done.
To get the students thinking about properties I ask, "If we switch the order in the product would it change the result?" I have students justify their reasoning. During the reasoning I remind students that the trigonometric form is just a different way to write a real number. For real numbers order of the factors does not change the result.
I now give the students the division part of the question. After finding the result I ask "If we switch the numerator and denominator will the result be the same?" Some students may have already stated that division will be different. I will ask "What will change in division?"
The questioning at this point is helping students see the importance of order and how different representations of a number do not change the outcome of an operation.
As class comes to an end I give students the following problems as homework:
From Larson's "Precalculus with Limits, 2nd ed." p. 476-477 #47, 50, 54, 60
These problems give students some basic practice in find the product and quotient of complex numbers in trigonometric form.