#### Transcript Chapter 0

Chapter 6 Additional Topics in Trigonometry 6.3 Polar Coordinates Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 Objectives: • Plot points in the polar coordinate system. • Find multiple sets of polar coordinates for a given point. • Convert a point from polar to rectangular coordinates. • Convert a point from rectangular to polar coordinates. • Convert an equation from rectangular to polar coordinates. • Convert an equation from polar to rectangular coordinates. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 Plotting Points in the Polar Coordinate System The foundation of the polar coordinate system is a horizontal ray that extends to the right. The ray is called the polar axis. The endpoint of the ray is called the pole. A point P in the polar coordinate system is represented by an ordered pair of numbers (r , ). We refer to the ordered pair P (r , ) as the polar coordinates of P. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 3 Plotting Points in the Polar Coordinate System (continued) We refer to the ordered pair P (r , ) as the polar coordinates of P. r is a directed distance from the pole to P. is an angle from the polar axis to the line segment from the pole to P. This angle can be measured in degrees or radians. Positive angles are measured counterclockwise from the polar axis. Negative angles are measured clockwise from the polar axis. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 4 Example: Plotting Points in a Polar Coordinate System Plot the point with the following polar coordinates: (3,315) Because 315° is a positive angle, draw 315 counterclockwise from the polar axis. Because r = 3 is positive, plot the point going out three units on the terminal side of Copyright © 2014, 2010, 2007 Pearson Education, Inc. (3,315) 315 5 Example: Plotting Points in a Polar Coordinate System Plot the point with the following polar coordinates: ( 2, ) Because π is a positive angle, draw counterclockwise from the polar axis. 2, Because r = –2 is negative, plot the point going out two units along the ray opposite the terminal side of Copyright © 2014, 2010, 2007 Pearson Education, Inc. 6 Example: Plotting Points in a Polar Coordinate System Plot the point with the following polar coordinates: 1, 2 Because is a negative angle, 2 draw clockwise 2 1, from the polar axis. Because r = –1 is negative, plot the point going out one unit along the ray opposite the terminal side of Copyright © 2014, 2010, 2007 Pearson Education, Inc. 2 2 7 Example: Finding Other Polar Coordinates for a Given Point Find another representation of 5, in which 4 r is positive and 2 4 . We want r positive and 2 4 . Add 2π to the angle and do not change r. 5, 5, 2 4 4 5, 4 8 9 5, 5, 4 4 4 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 8 Example: Finding Other Polar Coordinates for a Given Point Find another representation of 5, in which 4 r is negative and 0 2 . We want r negative and 0 2 . Add π to the angle and replace r with –r. 5, 5, 4 4 5, 4 4 5 5, 5, 4 4 4 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 9 Example: Finding Other Polar Coordinates for a Given Point Find another representation of 5, in which 4 r is positive and 2 0. We want r positive and 2 0. Subtract 2π from the angle and do not change r. 5, 5, 2 4 4 5, 4 7 8 5, 5, 4 4 4 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 10 Relations between Polar and Rectangular Coordinates Copyright © 2014, 2010, 2007 Pearson Education, Inc. 11 Point Conversion from Polar to Rectangular Coordinates To convert a point from polar coordinates ( r , ) to rectangular coordinates (x, y), use the formulas x r cos and y r sin . Copyright © 2014, 2010, 2007 Pearson Education, Inc. 12 Example: Polar-to-Rectangular Point Conversion Find the rectangular coordinates of the point with the following polar coordinates: 3, (r , ) 3, x r cos 3cos 3(1) 3 y r sin 3sin 3 0 0 The rectangular coordinates of 3, are (–3, 0). Copyright © 2014, 2010, 2007 Pearson Education, Inc. 13 Example: Polar-to-Rectangular Point Conversion Find the rectangular coordinates of the point with the following polar coordinates: 10, 6 (r , ) 10, 6 3 x r cos 10cos 10 5 3 6 2 10 1 5 y r sin 10sin 2 6 10, The rectangular coordinates of are 5 3, 5 . 6 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 14 Point Conversion from Rectangular to Polar Coordinates Copyright © 2014, 2010, 2007 Pearson Education, Inc. 15 Example: Rectangular-to-Polar Point Conversion Find polar coordinates of the point whose rectangular coordinates are 1, 3 . Step 1 Plot the point (x, y). Step 2 Find r by computing the distance from the origin to (x, y). r x2 y 2 y 3 (1) 3 2 x 1 2 1 3 4 2 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1, 3 16 Example: Rectangular-to-Polar Point Conversion Find polar coordinates of the point whose rectangular coordinates are 1, 3 . y Step 3 Find using tan x with the terminal side of passing through (x, y). y 3 3 tan 1 x x 1 5 3 y 3 1, 3 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 17 Example: Rectangular-to-Polar Point Conversion Find polar coordinates of the point whose rectangular coordinates are 1, 3 . One representation of 1, 3 . in polar coordinates is 5 (r , ) 2, . 3 x 1 y 3 1, 3 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 18 Equation Conversion from Rectangular to Polar Coordinates A polar equation is an equation whose variables are r and To convert a rectangular equation in x and y to a polar equation in r and replace x with r cos and y with r sin Copyright © 2014, 2010, 2007 Pearson Education, Inc. 19 Example: Converting Equations from Rectangular to Polar Coordinates Convert the following rectangular equation to a polar equation that expresses r in terms of 3x y 6 The polar equation for 3r cos r sin 6 is 3x y 6 r (3cos sin ) 6 6 r 3cos sin 6 r 3cos sin Copyright © 2014, 2010, 2007 Pearson Education, Inc. 20 Example: Converting Equations from Rectangular to Polar Coordinates Convert the following rectangular equation to a polar equation that expresses r in terms of r (r 2sin ) 0 x 2 ( y 1)2 1 2 2 r 0 r 2sin 0 r cos r sin 1 1 r 2sin r 2 cos 2 r 2 sin 2 2r sin 1 1 r 2 (cos2 sin 2 ) 2r sin 1 1 r 2 (1) 2r sin 1 1 r 2 2r sin 1 1 r 2 2r sin 0 The polar equation for x 2 ( y 1)2 1 is r 2sin Copyright © 2014, 2010, 2007 Pearson Education, Inc. 21 Equation Conversion from Polar to Rectangular Coordinates When we convert an equation from polar to rectangular coordinates, our goal is to obtain an equation in which the variables are x and y rather than r and . We use one or more of the following equations: y 2 2 2 r cos x r sin y r x y tan x To use these equations, it is sometimes necessary to square both sides, to use an identity, to take the tangent of both sides, or to multiply both sides by r. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 22 Example: Converting Equations from Polar to Rectangular Form Convert the polar equation to a rectangular equation in x and y: r4 2 r 2 x2 y 2 r4 r 2 16 r 4 or x2 y 2 16 0 x y 16 2 2 3 2 The rectangular equation for r 4 is x2 y 2 16. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 23 Example: Converting Equations from Polar to Rectangular Form Convert the polar equation to a rectangular equation in x 3 3 2 and y: 4 4 4 3 4 3 tan tan 4 tan 1 y tan x y 1 x y x 3 or y x 4 0 5 4 3 2 7 4 3 The rectangular equation for is y x. 4 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 24 Example: Converting Equations from Polar to Rectangular Form Convert the polar equation to a rectangular equation in x and y: r 2sec 2 3 r 2sec 4 4 2 r cos r cos 2 x 2 r 2sec or x 2 0 5 4 3 2 7 4 The rectangular equation for r 2sec is x 2. Copyright © 2014, 2010, 2007 Pearson Education, Inc. 25 Example: Converting Equations from Polar to Rectangular Form Convert the polar equation to a rectangular equation in x and y: r 10sin r 10sin x 2 y 2 10 y 0 r 2 r (10sin ) x 2 y 2 10 y 25 25 r 2 10r sin x2 ( y 5)2 25 r sin y The rectangular equation for r 10sin r 2 10 y is x2 ( y 5)2 25. 2 2 2 r x y 2 2 x y 10 y Copyright © 2014, 2010, 2007 Pearson Education, Inc. 26 Example: Converting Equations from Polar to Rectangular Form (continued) 2 Convert the polar equation to a rectangular equation in x and y: r 10sin The rectangular equation for r 10sin is x2 ( y 5)2 25. r 10sin or x2 ( y 5)2 25 3 4 4 (0,5) 0 5 4 Copyright © 2014, 2010, 2007 Pearson Education, Inc. 7 4 3 2 27