PRECALCULUS:GRAPHICAL,...-NASTA ED.
PRECALCULUS:GRAPHICAL,...-NASTA ED.
10th Edition
ISBN: 9780134672090
Author: Demana
Publisher: PEARSON
Expert Solution & Answer
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Chapter 6, Problem 23RE
Solution

a)

To find: the polar coordinates of a point P=5,0 that satisfy 0θ2π .

The polar coordinates for the point 5,0 that satisfy 0θ2π, are 5,0° , 5,π and 5,2π .

Given information:

The rectangular coordinates of the point P are 5,0 and 0θ2π .

Formula used:

  • Conversion of rectangular coordinates to polar coordinates:
  • Let the point P have rectangular coordinates x,y . The polar coordinates r,θ of P are computed by the following equations:

      r=±x2+y2, and θ=tan1yx+nπ, where n is an integer

  • The coordinates r,θ, r,θ+2nπ , and r,θ+π all the same point.

Calculation:

Substitute 5 for x and 0 for y in the equations: r=±x2+y2, and θ=tan1yx+nπ, where n is an integer.

  r=±52+02=±25=±5

Thus, r=±5

And

  θ=tan10/5+nπ=tan10+nπ=nπ

Given 0θ2π, then possible values for θ are 0,

  π and 2π .

Thus, the polar coordinates for the point 5,0 that satisfy 0θ2π, are 5,0° or 5,2π , and 5,π .

b)

To find: the polar coordinates of point P=5,0 that satisfy πθπ .

The polar coordinates for the point 5,0 that satisfy πθπ, are 5,0° , 5,π , and 5,π .

Given information:

The rectangular coordinates of the point P are 5,0 and πθπ .

Formula used:

  • Conversion of rectangular coordinates to polar coordinates:
  • Let the point P have rectangular coordinates x,y . The polar coordinates r,θ of P are computed by following equations:

      r=±x2+y2, and θ=tan1yx+nπ, where n is an integer.

  • The coordinates r,θ, r,θ+2nπ , and r,θ+π all the same point.

Calculation:

As calculated in part (a), r=±5 and θ=nπ , where n is the integer.

Given πθπ, then possible values for θ are π,

  0, and π .

Thus, the polar coordinates for the point 5,0 that satisfy πθπ, are 5,0° , 5,π, and 5,π .

c)

To find: the polar coordinates of point P=5,0 that satisfy 0θ4π .

The polar coordinates for the point 5,0 that satisfy 0θ4π, are 5,0°,

  5,2π,

  5,4π,

  5,π, and 5,3π .

Given information:

The rectangular coordinates of the point P are 5,0 and 0θ4π .

Formula used:

  • Conversion of rectangular coordinates to polar coordinates:
  • Let the point P have rectangular coordinates x,y . The polar coordinates r,θ of P are computed by following equations:

      r=±x2+y2, and θ=tan1yx+nπ, where n is an integer.

  • The coordinates r,θ, r,θ+2nπ , and r,θ+π all the same point.

Calculation:

As calculated in part (a), r=±5 and θ=nπ , where n is the integer.

Given 0θ4π, the possible values for θ are 0, π, 2π, 3π and 4π .

Thus, the polar coordinates for the point 5,0 that satisfy 0θ4π, are 5,0°, 5,2π, 5,4π, 5,π, and 5,3π .

Chapter 6 Solutions

PRECALCULUS:GRAPHICAL,...-NASTA ED.

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Prob. 57ECh. 6.5 - Prob. 58ECh. 6.5 - Prob. 59ECh. 6.5 - Prob. 60ECh. 6.5 - Prob. 61ECh. 6.5 - Prob. 62ECh. 6.5 - Prob. 63ECh. 6.5 - Prob. 64ECh. 6.5 - Prob. 65ECh. 6.5 - Prob. 66ECh. 6.5 - Prob. 67ECh. 6.5 - Prob. 68ECh. 6.5 - Prob. 69ECh. 6.5 - Prob. 70ECh. 6.5 - Prob. 71ECh. 6.5 - Prob. 72ECh. 6.5 - Prob. 73ECh. 6.6 - Prob. 1QRCh. 6.6 - Prob. 2QRCh. 6.6 - Prob. 3QRCh. 6.6 - Prob. 4QRCh. 6.6 - Prob. 5QRCh. 6.6 - Prob. 6QRCh. 6.6 - Prob. 7QRCh. 6.6 - Prob. 8QRCh. 6.6 - Prob. 9QRCh. 6.6 - Prob. 10QRCh. 6.6 - Prob. 1ECh. 6.6 - Prob. 2ECh. 6.6 - Prob. 3ECh. 6.6 - Prob. 4ECh. 6.6 - Prob. 5ECh. 6.6 - Prob. 6ECh. 6.6 - Prob. 7ECh. 6.6 - Prob. 8ECh. 6.6 - Prob. 9ECh. 6.6 - Prob. 10ECh. 6.6 - Prob. 11ECh. 6.6 - Prob. 12ECh. 6.6 - Prob. 13ECh. 6.6 - Prob. 14ECh. 6.6 - Prob. 15ECh. 6.6 - Prob. 16ECh. 6.6 - Prob. 17ECh. 6.6 - Prob. 18ECh. 6.6 - Prob. 21ECh. 6.6 - Prob. 22ECh. 6.6 - Prob. 23ECh. 6.6 - Prob. 24ECh. 6.6 - Prob. 25ECh. 6.6 - Prob. 26ECh. 6.6 - Prob. 27ECh. 6.6 - Prob. 28ECh. 6.6 - Prob. 29ECh. 6.6 - 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