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2. Using Boundless Presentations Boundless Teaching Platform Boundless empowers educators to engage their students with affordable, customizable textbooks and intuitive teaching tools. The free Boundless Teaching Platform gives educators the ability to customize textbooks in more than 20 subjects that align to hundreds of popular titles. Get started by using high quality Boundless books, or make switching to our platform easier by building from Boundless content pre-organized to match the assigned textbook. This platform gives educators the tools they need to assign readings and assessments, monitor student activity, and lead their classes with pre-made teaching resources. Get started now at: • The Appendix The appendix is for you to use to add depth and breadth to your lectures. You can simply drag and drop slides from the appendix into the main presentation to make for a richer lecture experience. http://boundless.com/teaching-platform • Free to edit, share, and copy Feel free to edit, share, and make as many copies of the Boundless presentations as you like. We encourage you to take these presentations and make them your own. If you have any questions or problems please email: educators@boundless.com Free to share, print, make copies and changes. Get yours at www.boundless.com

3. About Boundless • Boundless is an innovative technology company making education more affordable and accessible for students everywhere. The company creates the world’s best open educational content in 20+ subjects that align to more than 1,000 popular college textbooks. Boundless integrates learning technology into all its premium books to help students study more efficiently at a fraction of the cost of traditional textbooks. The company also empowers educators to engage their students more effectively through customizable books and intuitive teaching tools as part of the Boundless Teaching Platform. More than 2 million learners access Boundless free and premium content each month across the company’s wide distribution platforms, including its website, iOS apps, Kindle books, and iBooks. To get started learning or teaching with Boundless, visit boundless.com. Free to share, print, make copies and changes. Get yours at www.boundless.com

4. The Polar Coordinate System Complex Numbers and Polar Coordinates Complex Numbers ] Complex Numbers and Polar Coordinates Free to share, print, make copies and changes. Get yours at www.boundless.com

5. Complex Numbers and Polar Coordinates > The Polar Coordinate System The Polar Coordinate System • Introduction to the Polar Coordinate System • Converting Between Polar and Cartesian Coordinates • Conics in Polar Coordinates • Other Curves in Polar Coordinates Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/complex-numbers-and-polar-coordinates-344/the-polar-coordinate-system-417/

6. Complex Numbers and Polar Coordinates > Complex Numbers Complex Numbers • Introduction to Complex Numbers • Addition and Subtraction of Complex Numbers • Multiplication of Complex Numbers • Complex Numbers and the Binomial Theorem • Complex Conjugates • Division of Complex Numbers • Complex Numbers in Polar Coordinates Free to share, print, make copies and changes. Get yours at www.boundless.com www.boundless.com/algebra/textbooks/boundless-algebra-textbook/complex-numbers-and-polar-coordinates-344/complex-numbers-351/

7. Appendix Free to share, print, make copies and changes. Get yours at www.boundless.com

8. Complex Numbers and Polar Coordinates Key terms • angular coordinateAn angle measured from the polar axis, usually counter-clockwise. • Archimedes’ spiralA curve given by an equation of the form • complexa number, of the form , where and are real numbers and is the square root of . • complex conjugateFor the number , this is . • directrixA fixed line used to described a curve. • eccentricityA measure of deviation from a prescribed curve. • imaginary numbera number of the form , where is a real number and the imaginary unit • modulusThe length of a complex number, • polar axisA ray from the pole in the reference direction. • poleThe reference point of the polar graph. • radiusA distance measured from the pole. • real numberAn element of the set of real numbers. The set of real numbers include the rational numbers and the irrational numbers, but not all complex numbers. Free to share, print, make copies and changes. Get yours at www.boundless.com

9. Complex Numbers and Polar Coordinates • rose curveA curve given by an equation of the form or Free to share, print, make copies and changes. Get yours at www.boundless.com

10. Complex Numbers and Polar Coordinates Rose Curves Complex graphs generated by the simple polar formulas that generate rose curves: and  where . Free to share, print, make copies and changes. Get yours at www.boundless.com OpenStax."CNX_Precalc_Figure_08_04_016new.jpg."CC BY-SA 3.0http://cnx.org/contents/E6wQevFf@5.241:pj7U1aMb@6/Polar-Coordinates-GraphsView on Boundless.com

11. Complex Numbers and Polar Coordinates The parameters for polar coordinates A point in the first quadrant with Cartesian coordinates and polar coordinates . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Complex_number_illustration_modarg.svg."Free Art Licensehttps://en.wikipedia.org/wiki/Complex_number#Polar_formView on Boundless.com

12. Complex Numbers and Polar Coordinates The complex point The complex number is plotted in the complex plane, to the left on the real axis, and up on the imaginary axis. Free to share, print, make copies and changes. Get yours at www.boundless.com OpenStax."CNX_Precalc_Figure_03_01_002.jpg."License: Otherhttps://cnx.org/contents/9AFg5Sd0@11/Complex-NumbersView on Boundless.com

13. Complex Numbers and Polar Coordinates The length of a complex number The length of the line segment from the origin to the point  is .  This comes from the Pythagorean Theorem. Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."440px-Complex_number_illustration.svg.png."CC0 1.0 Universalhttps://en.wikipedia.org/wiki/Complex_numberView on Boundless.com

14. Complex Numbers and Polar Coordinates Trigonometry Right Triangle A right triangle with rectangular (Cartesian) coordinates and equivalent polar coordinates. Free to share, print, make copies and changes. Get yours at www.boundless.com OpenStax."CNX_Precalc_Figure_08_03_010new.jpg."CC BY-SA 3.0http://cnx.org/contents/E6wQevFf@5.241:weepUiqV@7/Polar-CoordinatesView on Boundless.com

15. Complex Numbers and Polar Coordinates Trigonometry Right Triangle A right triangle with rectangular (Cartesian) coordinates and equivalent polar coordinates. Free to share, print, make copies and changes. Get yours at www.boundless.com OpenStax."CNX_Precalc_Figure_08_03_007.jpg."CC BY-SA 3.0https://cnx.org/contents/weepUiqV@7/Polar-CoordinatesView on Boundless.com

16. Complex Numbers and Polar Coordinates Archimedes' Spiral The formula that generates the graph of a spiral is  for . Free to share, print, make copies and changes. Get yours at www.boundless.com OpenStax."CNX_Precalc_Figure_08_04_020new.jpg."CC BY-SA 3.0http://cnx.org/contents/E6wQevFf@5.241:pj7U1aMb@6/Polar-Coordinates-GraphsView on Boundless.com

17. Complex Numbers and Polar Coordinates Parts of a Parabola Consider the parabola . Any conic may be determined by three characteristics: a single focus, a fixed line called the directrix, and the ratio of the distances of each to a point on the graph.   Free to share, print, make copies and changes. Get yours at www.boundless.com OpenStax."CNX_Precalc_Figure_10_05_001.jpg."CC BY-SA 3.0http://cnx.org/contents/E6wQevFf@5.241:F2cubBPi@6/Conic-Sections-in-Polar-CoordiView on Boundless.com

18. Complex Numbers and Polar Coordinates Polar and Coordinate Grid of Equivalent Points The rectangular coordinate  is the same as the polar coordinate  as plotted on the two grids above. Free to share, print, make copies and changes. Get yours at www.boundless.com OpenStax."CNX_Precalc_Figure_08_03_008.jpg."CC BY-SA 3.0http://cnx.org/contents/E6wQevFf@5.241:weepUiqV@7/Polar-CoordinatesView on Boundless.com

19. Complex Numbers and Polar Coordinates Plotting a point on a Polar Grid Plot of the point ,by moving  units in the counterclockwise direction and then a length of  from the pole. Free to share, print, make copies and changes. Get yours at www.boundless.com OpenStax."CNX_Precalc_Figure_08_03_002.jpg."CC BY-SA 3.0https://cnx.org/contents/weepUiqV@7/Polar-CoordinatesView on Boundless.com

20. Complex Numbers and Polar Coordinates Polar Graph Paper A polar grid with several angles labeled in degrees Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."250px-Polar_graph_paper.svg.png."CC BY-SA 3.0https://en.wikipedia.org/wiki/Polar_coordinate_systemView on Boundless.com

21. Complex Numbers and Polar Coordinates Examples of Polar Coordinates Points in the polar coordinate system with pole  and polar axis . In green, the point with radial coordinate and angular coordinate degrees or . In blue, the point . Free to share, print, make copies and changes. Get yours at www.boundless.com Wikipedia."Examples_of_Polar_Coordinates.svg.png."CC BY-SA 3.0https://en.wikipedia.org/wiki/Polar_coordinate_systemView on Boundless.com

22. Complex Numbers and Polar Coordinates Attribution • Wiktionary."real number."CC BY-SA 3.0http://en.wiktionary.org/wiki/real_number • Wiktionary."imaginary number."CC BY-SA 3.0http://en.wiktionary.org/wiki/imaginary_number • Wiktionary."complex."CC BY-SA 3.0http://en.wiktionary.org/wiki/complex • Wikipedia."Complex number."CC BY-SA 3.0http://en.wikipedia.org/wiki/Complex_number • Wikipedia."Polar Coordinate system."CC BY-SA 3.0https://en.wikipedia.org/wiki/Polar_coordinate_system • OpenStax."Polar Coordinates."CC BY-SA 3.0https://cnx.org/contents/weepUiqV@7/Polar-Coordinates • Wikipedia."Polar coordinate system."CC BY-SA 3.0https://en.wikipedia.org/wiki/Polar_coordinate_system#Converting_between_polar_and_Cartesian_coordinates • OpenStax."Polar Coordinates."CC BY-SA 3.0http://cnx.org/contents/E6wQevFf@5.241:weepUiqV@7/Polar-Coordinates • Wikipedia."Polar Coordinate System."CC BY-SA 3.0https://en.wikipedia.org/wiki/Polar_coordinate_system#Converting_between_polar_and_Cartesian_coordinates • OpenStax."Polar Coordinates:Graphs."CC BY-SA 3.0http://cnx.org/contents/E6wQevFf@5.241:pj7U1aMb@6/Polar-Coordinates-Graphs • OpenStax."Conic Sections in Polar Coordinates."CC BY-SA 3.0http://cnx.org/contents/E6wQevFf@5.241:F2cubBPi@6/Conic-Sections-in-Polar-Coordi • Wikipedia."Polar Coordinate system."CC BY-SA 3.0https://en.wikipedia.org/wiki/Polar_coordinate_system#Polar_equation_of_a_curve • OpenStax."Polar Coordinates: Graphs."CC BY-SA 3.0http://cnx.org/contents/E6wQevFf@5.241:pj7U1aMb@6/Polar-Coordinates-Graphs Free to share, print, make copies and changes. Get yours at www.boundless.com