9.6 Polar Coordinates Digital Lesson

HWQ 3/24 Find a set of parametric equations to represent the graph of using the parameter. Sketch a graph on showing orientation. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 t012 x y y x 1 1 2

One way to give someone directions is to tell them to go three blocks East and five blocks South. Another way to give directions is to point and say “Go a half mile in that direction.” Polar graphing is like the second method of giving directions. Each point is determined by a distance and an angle. Initial ray A polar coordinate pair determines the location of a point. 9.6 Polar Coordinates

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 The polar coordinate system is formed by fixing a point, O, which is the pole (or origin). Definition: Polar Coordinate System  = directed angle Polar axis r = directed distance O Pole (Origin) The polar axis is the ray constructed from O. Each point P in the plane can be assigned polar coordinates (r,  ). P = (r,  ) r is the directed distance from O to P.  is the directed angle (counterclockwise) from the polar axis to OP.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Plotting Points a) The point lies two units from the pole on the terminal side of the angle Plotting Points

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Plotting Points units from the pole Plotting Points b) Plot the point

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Multiple Representations of Points There are many ways to represent the point additional ways to represent the point

Plot the polar point, then find one additional representation. Copyright © by Houghton Mifflin Company, Inc. All rights reserved

9 Polar and Rectangular Coordinate (r,  ) (x, y)  Pole x y (Origin) y r x The relationship between rectangular and polar coordinates is as follows. The point (x, y) lies on a circle of radius r, therefore, r 2 = x 2 + y 2. Definitions of trigonometric functions

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Example: Coordinate Conversion Coordinate Conversion (Pythagorean Identity) Example: Convert the point into rectangular coordinates. x = r cos(θ) y = r sin(θ) r 2 = x 2 + y 2 tan(θ) = y/x

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Example: Convert the point into rectangular coordinates. Example: Convert the point into rectangular coordinates. Example: Convert the point into rectangular coordinates. x = r cos(θ) y = r sin(θ)

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Example: Coordinate Conversion Example: Convert the point (1,1) into polar coordinates.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 Example: Convert the point (0,2) into polar coordinates. Example: Convert the point (-3,4) into polar coordinates. r 2 = x 2 + y 2 tan(θ) = y/x

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Example: Coordinate Conversion Equation Conversion x = r cos(θ) y = r sin(θ) r 2 = x 2 + y 2 tan(θ) = y/x To convert a rectangular equation to polar form, use: To convert a polar equation to rectangular form, use:

Equation Conversion ExampleFor the rectangular equation 3x + 2y = 4, (a)convert to a polar equation, (b)use a graphing calculator to graph the rectangular equation, and (c)use a graphing calculator to graph the polar equation for 0 °    360 °. Solution (a)Let x = r cos  and y = r sin  to get

Converting a Rectangular Equation to Polar Form (b)Solve the rectangular equation for y to get (c)

Converting a Polar Equation to Rectangular Form ExampleFor the polar equation (a)convert to a rectangular equation, (b)use a graphing calculator to graph the polar equation for 0    2 , and (c)use a graphing calculator to graph the rectangular equation. Solution:Multiply both sides by the denominator.

Converting a Polar Equation to Rectangular Form Square both sides. Rectangular equation

Converting a Polar Equation to Rectangular Form (b)The figure shows a graph with polar coordinates. (c)Solving x 2 = –8(y – 2) for y, we obtain

Homework Day pg. 680: 1, 5, 9, 13, 17, 21, 25, 29, 33, odds Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20

9.6 Polar Coordinates Day 2 Digital Lesson Answer equation conversion problems at end of lesson.

HWQ 3/25 Find 2 different sets of polar coordinates for the rectangular point: Exact values only. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 23 Example: Coordinate Conversion 9.6 Day 2 - More Equation Conversion x = r cos(θ) y = r sin(θ) r 2 = x 2 + y 2 tan(θ) = y/x To convert a rectangular equation to polar form, use: To convert a polar equation to rectangular form, use:

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 24 Converting Equations Ex: Convert the rectangular equation to polar form:

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 25 Ex: Convert the rectangular equation to polar form:

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 26 Ex: Convert the rectangular equation to polar form:

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 27 Example: Converting Polar Equations to Rectangular Equation Conversion Convert the polar equation into a rectangular equation. Multiply each side by r. Substitute rectangular coordinates. Equation of a circle with center (0, 2) and radius of 2 Polar form

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 28 Example: Convert the polar equation r = 2 to rectangular form.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 29 Example: Convert the polar equation to rectangular form.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 30 Example: Convert the polar equation to rectangular form. x = r cos(θ) y = r sin(θ)

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 31 Example: Convert from rectangular to polar form:

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 32 Example: Convert the polar equation to rectangular form. x = r cos(θ) y = r sin(θ)

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 33 Example: Convert the polar equation to rectangular form. x = r cos(θ) y = r sin(θ)

Homework Day pg , 7, 11, 15, 19, 23, 27, 31, 35, odds Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 34

HWQ 3/26 Convert the rectangular equation to polar form: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 35