Complex Numbers, Polar Equations, and Parametric Equations. Copyright 2017, 2013, 2009 Pearson Education, Inc.
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1 8 Complex Numbers, Polar Equations, and Parametric Equations Copyright 2017, 2013, 2009 Pearson Education, Inc. 1
2 8.5 Polar Equations and Graphs Polar Coordinate System Graphs of Polar Equations Conversion from Polar to Rectangular Equations Classification of Polar Equations Copyright 2017, 2013, 2009 Pearson Education, Inc. 2
3 Polar Coordinate System The polar coordinate system is based on a point, called the pole, and a ray, called the polar axis. Copyright 2017, 2013, 2009 Pearson Education, Inc. 3
4 Polar Coordinate System Point P has rectangular coordinates (x, y). Point P can also be located by giving the directed angle θ from the positive x-axis to ray OP and the directed distance r from the pole to point P. The polar coordinates of point P are (r, θ). Copyright 2017, 2013, 2009 Pearson Education, Inc. 4
5 Polar Coordinate System If r > 0, then point P lies on the terminal side of θ. If r < 0, then point P lies on the ray pointing in the opposite direction of the terminal side of θ, a distance r from the pole. Copyright 2017, 2013, 2009 Pearson Education, Inc. 5
6 Rectangular and Polar Coordinates If a point has rectangular coordinates (x, y) and polar coordinates (r, ), then these coordinates are related as follows. Copyright 2017, 2013, 2009 Pearson Education, Inc. 6
7 Example 1 PLOTTING POINTS WITH POLAR COORDINATES Plot each point by hand in the polar coordinate system. Then, determine the rectangular coordinates of each point. (a) P(2, 30 ) r = 2 and θ = 30, so P is located 2 units from the origin in the positive direction on a ray making a 30 angle with the polar axis. Copyright 2017, 2013, 2009 Pearson Education, Inc. 7
8 Example 1 PLOTTING POINTS WITH POLAR COORDINATES (continued) Using the conversion formulas: The rectangular coordinates are Copyright 2017, 2013, 2009 Pearson Education, Inc. 8
9 Example 1 PLOTTING POINTS WITH POLAR COORDINATES (continued) Since r is negative, Q is 4 units in the opposite direction from the pole on an extension of the ray. The rectangular coordinates are Copyright 2017, 2013, 2009 Pearson Education, Inc. 9
10 Example 1 PLOTTING POINTS WITH POLAR COORDINATES (continued) Since θ is negative, the angle is measured in the clockwise direction. The rectangular coordinates are Copyright 2017, 2013, 2009 Pearson Education, Inc. 10
11 Note While a given point in the plane can have only one pair of rectangular coordinates, this same point can have an infinite number of pairs of polar coordinates. Copyright 2017, 2013, 2009 Pearson Education, Inc. 11
12 Example 2 GIVING ALTERNATIVE FORMS FOR COORDINATES OF A POINT (a) Give three other pairs of polar coordinates for the point P(3, 140 ). Three pairs of polar coordinates for the point P(3, 140 ) are (3, 220 ), ( 3, 320 ), and ( 3, 40 ). Copyright 2017, 2013, 2009 Pearson Education, Inc. 12
13 Example 2 GIVING ALTERNATIVE FORMS FOR COORDINATES OF A POINT (continued) (b) Determine two pairs of polar coordinates for the point with the rectangular coordinates ( 1, 1). The point ( 1, 1) lies in quadrant II. Since value for θ is 135. one possible Two pairs of polar coordinates are Copyright 2017, 2013, 2009 Pearson Education, Inc. 13
14 Graphs of Polar Equations An equation in which r and θ are the variables is a polar equation. Define the polar equation of the line ax + by = c as follows: Convert from rectangular to polar coordinates. Factor out r. General form for the polar equation of a line Copyright 2017, 2013, 2009 Pearson Education, Inc. 14
15 Graphs of Polar Equations Define the polar equation of the circle x 2 + y 2 = a 2 as follows: General form for the polar equation of a circle Copyright 2017, 2013, 2009 Pearson Education, Inc. 15
16 Example 3 FINDING POLAR EQUATIONS OF LINES AND CIRCLES For each rectangular equation, give the equivalent polar equation and sketch its graph. (a) y = x 3 This is the equation of a line. Rewrite y = x 3 in standard form. x y 3 r cos r sin 3 r(cos sin ) 3 y = x 3 is equivalent to r 3 cos sin Copyright 2017, 2013, 2009 Pearson Education, Inc. 16
17 Example 3 FINDING POLAR EQUATIONS OF LINES AND CIRCLES (continued) y x 3 (rectangular) r 3 cos sin (polar) Copyright 2017, 2013, 2009 Pearson Education, Inc. 17
18 Example 3 FINDING POLAR EQUATIONS OF LINES AND CIRCLES (continued) (b) This is the graph of a circle with center at the origin and radius 2. Note that in polar coordinates it is possible for r < 0. The graphs of r = 2 and r = 2 coincide. Copyright 2017, 2013, 2009 Pearson Education, Inc. 18
19 Example 3 FINDING POLAR EQUATIONS OF LINES AND CIRCLES (continued) x y 2 2 r 2 (polar) 4 (rectangular) Copyright 2017, 2013, 2009 Pearson Education, Inc. 19
20 Example 4 GRAPHING A POLAR EQUATION (CARDIOID) Find some ordered pairs to determine a pattern of values of r. Copyright 2017, 2013, 2009 Pearson Education, Inc. 20
21 Example 4 GRAPHING A POLAR EQUATION (CARDIOID) Connect the points in order from (2, 0 ) to (1.9, 30 ) to (1.7, 45 ) and so on. This curve is called a cardioid. The curve has been graphed on a polar grid. Copyright 2017, 2013, 2009 Pearson Education, Inc. 21
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22 Example 4 GRAPHING A POLAR EQUATION (CARDIOD) (continued) Choose degree mode and graph values of θ in the interval [0, 360 ]. Copyright 2017, 2013, 2009 Pearson Education, Inc. 22
23 Example 5 GRAPHING A POLAR EQUATION (ROSE) Find some ordered pairs to determine a pattern of values of r. Copyright 2017, 2013, 2009 Pearson Education, Inc. 23
24 Example 5 GRAPHING A POLAR EQUATION (ROSE) (continued) Plot the points from the table in order gives the graph of a four-leaved rose. Note how the graph is developed with a continuous curve, starting with the upper half of the right horizontal leaf and ending with the lower half of that leaf. Copyright 2017, 2013, 2009 Pearson Education, Inc. 24
25 Example 6 GRAPHING A POLAR EQUATION (LEMNISCATE) Graph. Complete a table of ordered pairs, and sketch the graph. Values of θ for 45 θ 135 are not included in the table because the corresponding values of 2θ are negative. Values of θ larger than 180 give 2θ larger than 360 and would repeat the values already found. Copyright 2017, 2013, 2009 Pearson Education, Inc. 25
26 Example 6 GRAPHING A POLAR EQUATION (LEMNISCATE) (continued) Copyright 2017, 2013, 2009 Pearson Education, Inc. 26
27 Example 6 To graph GRAPHING A POLAR EQUATION (LEMNISCATE) (continued) with a graphing calculator, let Copyright 2017, 2013, 2009 Pearson Education, Inc. 27
28 Example 7 GRAPHING A POLAR EQUATION (SPIRAL OF ARCHIMEDES) Graph r = 2θ, (θ measured in radians). Find some ordered pairs to determine a pattern of values of r. Since r = 2θ, also consider negative values of θ. Copyright 2017, 2013, 2009 Pearson Education, Inc. 28
29 Example 7 GRAPHING A POLAR EQUATION (SPIRAL OF ARCHIMEDES) (continued) Copyright 2017, 2013, 2009 Pearson Education, Inc. 29
30 Example 8 CONVERTING A POLAR EQUATION TO A RECTANGULAR EQUATION Convert the equation coordinates, and graph. to rectangular x r r sin 4 y y Multiply by 1 + sin θ. Square each side. Expand. Rectangular equation Copyright 2017, 2013, 2009 Pearson Education, Inc. 30
31 Example 8 CONVERTING A POLAR EQUATION TO A RECTANGULAR EQUATION (cont.) The graph is plotted with the calculator in polar mode. Copyright 2017, 2013, 2009 Pearson Education, Inc. 31
32 Classifying Polar Equations Circles and Lemniscates Copyright 2017, 2013, 2009 Pearson Education, Inc. 32
33 Classifying Polar Equations Limaçons Copyright 2017, 2013, 2009 Pearson Education, Inc. 33
34 Classifying Polar Equations 2n leaves if n is even, n 2 Rose Curves n leaves if n is odd Copyright 2017, 2013, 2009 Pearson Education, Inc. 34
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