Copyright © 2013, 2009, 2005 Pearson Education, Inc. 45 8 Complex Numbers, Polar Equations, and Parametric Equations Copyright © 2013, 2009, 2005 Pearson

Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1

8

Complex Numbers, Polar Equations, and Parametric Equations


Sections 8.5–8.6


8.5Polar Equations and Graphs

Complex Numbers, Polar Equations, and Parametric Equations8


Polar Equations and Graphs8.5Polar Coordinate System ▪ Graphs of Polar Equations ▪ Converting from Polar to Rectangular Equations ▪ Classifying Polar Equations


Plot each point by hand in the polar coordinate system. Then, determine the rectangular coordinates of each point.

8.5 Example 1 Plotting Points With Polar Coordinates

(page 380)

The rectangular coordinates

of P(4, 135°) are


8.5 Example 1 Plotting Points With Polar Coordinates (cont.)

The rectangular coordinates of


8.5 Example 1 Plotting Points With Polar Coordinates (cont.)

The rectangular coordinates of are (0, –2).


Give three other pairs of polar coordinates for the point P(5, –110°).

8.5 Example 2(a) Giving Alternative Forms for Coordinates of a Point (page 381)

Three pairs of polar coordinates for the point P(5, −110º) are (5, 250º), (−5, 70º), and (−5, −290º).

Other answers are possible.


Give two pairs of polar coordinates for the point with the rectangular coordinates

8.5 Example 2(b) Giving Alternative Forms for Coordinates of a Point (page 394)

The point lies in quadrant II.

Since , one possible value for θ is 300°.

Other answers are possible.

Two pairs of polar coordinates are (12, 300°) and (−12, 120°).


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 = 2x – 4

In standard form, the equation is 2x – y = 4, so a = 2, b = –1, and c = 4.

The general form for the polar equation of a line is

y = 2x – 4 is equivalent to


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.




8.5 Example 4 Graphing a Polar Equation (Cardioid)

(page 383)

Find some ordered pairs to determine a pattern of values of r.


8.5 Example 4 Graphing a Polar Equation (Cardioid) (cont.)

Connect the points in order from (1, 0°) to (.5, 30°) to (.1, 60°) and so on.


8.5 Example 4 Graphing a Polar Equation (Cardioid) (cont.)

Graphing calculator solution


8.5 Example 5 Graphing a Polar Equation (Rose) (page 384)



8.5 Example 5 Graphing a Polar Equation (Rose) (cont.)

Connect the points in order from (4, 0°) to (3.6, 10°) to (2.0, 20°) and so on.


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.




(cont.)

0

0±2.8±2.1 ±2.1±2.8

±2.8±2.1 ±2.1±2.8



(cont.)



(cont.) Graphing calculator solution


8.5 Example 7 Graphing a Polar Equation (Spiral of Archimedes) (page 385)

Graph r = –θ (θ measured in radians).



8.5 Example 7 Graphing a Polar Equation (Spiral of Archimedes) (cont.)


8.5 Example 8 Converting a Polar Equation to a Rectangular Equation (page 386)

Convert the equation to rectangular coordinates and graph.

Multiply both sides by 1 – cos θ.

Square both sides.


8.5 Example 8 Converting a Polar Equation to a Rectangular Equation (cont.)

The graph is a parabola with vertex at and axis y = 0.

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Copyright © 2013, 2009, 2005 Pearson Education, Inc. 45 8 Complex Numbers, Polar Equations, and Parametric Equations Copyright © 2013, 2009, 2005 Pearson