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# Complex Numbers and Trigonometry

Lesson 6 of 11

## Objective: SWBAT write a complex number in trigonometric form.

*45 minutes*

#### Bell work

*5 min*

Today students will write complex numbers in trigonometric form which is also called polar form. Since we have not discussed the polar coordinate system I use the term trigonometric form.

I begin by asking students to think about how to use trigonometry to write a complex number in trigonometric form. As students think, I ask questions focus questions such as:

- What is needed to use trigonometry?
- Would plotting the number help us determine how to do trigonometry?
- Could we use a process similar to finding the component form of vectors?

My goal for the bell work is to get students to see how they can find the angle rotated from the x axis and the absolute value of the complex number can be used to an appropriate form.

#### Resources

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Now that students have started to reason about how to writing trigonometric form. I continue with the bell problem. Students process through the bell work by plotting the point then drawing a triangle. with that they find the angle and the distance from the origin. From that they get a formula. As they do this I remind students how we write a vector using standard unit vectors.

Once students have used the bell problem to find a trigonometric form I move students to generalize the process.

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#### Converting to Complex form

*10 min*

Now that students have a general process for converting to trigonometric form, students work on problems. The first problem is looks very similar to the z=3+2i but the other 2 may confuse the students. The issue will be not thinking about the real or imaginary term equaling zero. I remind students to graph if number on the complex plane to help them understand what the numbers looks like. Once this is done students are able to determine the equation.

I move around the room checking results as the students work and helping students as needed. As I move around the room I also keep note of students who have correct answers and those that have done the problem differently. When I notice most students have completed a couple of the problems, I begin asking students with answers and unusual methods to put the answers on the board. We continue until all the solutions are shared with the class.

#### Resources

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I now have students determine how to convert form trigonometric form to standard complex form. I give students the page 2 of resource. I do not explain how to do the problems but let the students think about how they can find the standard form. As students discuss the problems I ask questions such as:

- What does standard complex form look like?(z=a+bi)
- What is a in the trigonometric form?
- How can we simplify the expression?

My goal is for students to see they just need to evaluate the trigonometric expression and multiply the result by r to determine the coefficient and constant in the standard form. Most students will see how to do this quickly while others will struggle since some numbers are represented using trigonometric functions.

In these problems I have used angles One issue that sometimes deal with is the tendency to use the calculator on angles that we have exact values (multiples of 30 degrees and 45 degrees). If students give me the calculator answer I will ask them to also find the exact answer. Any time I can give students time to practice with these angles I use.

#### Resources

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#### Closure

*5 min*

With about 5 minutes left in class I ask students to answer this question on an exit slip. I am hoping students see 2 issues with the example. First the coefficients on the terms are not the same and second the angles are not the same. Any students that do not see these 2 issues will need some clarification as we continue with complex numbers. I will review with the students on an individual basis during any work time we have.

#### Resources

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- UNIT 1: Introduction to Learning Mathematics
- UNIT 2: Functions and Piecewise Functions
- UNIT 3: Exponential and Logarithmic functions
- UNIT 4: Matrices
- UNIT 5: Conics
- UNIT 6: Solving Problems Involving Triangles
- UNIT 7: Trigonometry as a Real-Valued Functions
- UNIT 8: Graphing Trigonometric Functions
- UNIT 9: Trigonometric Identities
- UNIT 10: Solving Equations
- UNIT 11: Vectors and Complex Numbers
- UNIT 12: Parametric and Polar graphs and equations

- LESSON 1: Introduction to Vectors
- LESSON 2: Component Form of Vectors
- LESSON 3: Operation with Vectors
- LESSON 4: Solving Problems with Vectors
- LESSON 5: Review of Complex Numbers
- LESSON 6: Complex Numbers and Trigonometry
- LESSON 7: Operations of Complex Numbers in Trigonometric Form
- LESSON 8: DeMoivre's Theorem
- LESSON 9: Roots of Complex Numbers
- LESSON 10: Review
- LESSON 11: Assessment