8 Complex Numbers Polar Equations and Parametric Equations

8. 5 Polar Equations and Graphs Polar Coordinate System ▪ Graphs of Polar Equations ▪ Converting from Polar to Rectangular Equations ▪ Classifying Polar Equations Copyright © 2013, 2009, 2005 Pearson Education, Inc. 47

8. 5 Example 1 Plotting Points With Polar Coordinates (page 380) Plot each point by hand in the polar coordinate system. Then, determine the rectangular coordinates of each point. The rectangular coordinates of P(4, 135°) are Copyright © 2013, 2009, 2005 Pearson Education, Inc. 48

8. 5 Example 1 Plotting Points With Polar Coordinates (cont. ) The rectangular coordinates

8. 5 Example 2(a) Giving Alternative Forms for Coordinates of a Point (page 381) Give three other pairs of polar coordinates for the point P(5, – 110°). Three pairs of polar coordinates for the point P(5, − 110º) are (5, 250º), (− 5, 70º), and (− 5, − 290º). Other answers are possible. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 51

8. 5 Example 2(b) Giving Alternative Forms for Coordinates of a Point (page 394) Give two pairs of polar coordinates for the point with the rectangular coordinates The point Since 300°. lies in quadrant II. , one possible value for θ is Two pairs of polar coordinates are (12, 300°) and (− 12, 120°). Other answers are possible. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 52

8. 5 Example 3 Examining Polar and Rectangular Equations of Lines and Circles (page 382) For each rectangular equation, give the equivalent polar equation and sketch its graph. (a) y = 2 x – 4 In standard form, the equation is 2 x – y = 4, so a = 2, b = – 1, and c = 4. The general form for the polar equation of a line is y = 2 x – 4 is equivalent to Copyright © 2013, 2009, 2005 Pearson Education, Inc. 53

8. 5 Example 3 Examining Polar and Rectangular Equations of Lines and Circles (cont. ) This is the equation of a circle with center at the origin and radius 5. Note that in polar coordinates it is possible for r < 0. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 55

8. 5 Example 4 Graphing a Polar Equation (Cardioid) (page 383) Find some ordered

8. 5 Example 4 Graphing a Polar Equation (Cardioid) (cont. ) Connect the points

8. 5 Example 4 Graphing a Polar Equation (Cardioid) (cont. ) Graphing calculator solution

8. 5 Example 5 Graphing a Polar Equation (Rose) (page 384) Find some ordered

8. 5 Example 5 Graphing a Polar Equation (Rose) (cont. ) Connect the points

8. 5 Example 5 Graphing a Polar Equation (Rose) (cont. ) Graphing calculator solution

8. 5 Example 6 Graphing a Polar Equation (Lemniscate) (page 385) The graph only exists for [0°, 90°] and [180°, 270°] because sin 2θ must be positive. Find some ordered pairs to determine a pattern of values of r. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 63

8. 5 Example 6 Graphing a Polar Equation (Lemniscate) (cont. ) ± 2. 1

8. 5 Example 6 Graphing a Polar Equation (Lemniscate) (cont. ) Copyright © 2013,

8. 5 Example 6 Graphing a Polar Equation (Lemniscate) (cont. ) Graphing calculator solution

8. 5 Example 7 Graphing a Polar Equation (Spiral of Archimedes) (page 385) Graph r = –θ (θ measured in radians). Find some ordered pairs to determine a pattern of values of r. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 67

8. 5 Example 7 Graphing a Polar Equation (Spiral of Archimedes) (cont. ) Copyright

8. 5 Example 8 Converting a Polar Equation to a Rectangular Equation (page 386) Convert the equation coordinates and graph. to rectangular Multiply both sides by 1 – cos θ. Square both sides. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 69