Chapter 5 Complex Numbers, Polar Coordinates, and Parametric Equations 5.3 Polar Coordinates

Getting a New Perspective We have worked extensively in the Cartesian , plotting points, graphing equations, and using the properties of the Cartesian to investigate functions and solve problems. In this section, we introduce a different coordinate system that will give you a new perspective of the plane. In this system, points do not have unique ordered pairs associated with them and some complicated-looking graphs have very simple equations.

1 Plot 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 P in the polar coordinate system is represented by an ordered pair of numbers (풓, 휽). We refer to the ordered pair 푷 = (풓, 휽) as the polar coordinates of P.

 r is a directed from the pole to P. r can be positive, negative, or zero.  θ is an from the polar axis to the segment from the pole to P. This angle can be measured in degrees or . o Positive are measured counterclockwise from the polar axis. o Negative angles are measured clockwise from the polar axis.

The Sign of r and Point’s Location in Polar Coordinates The 푷 = (풓, 휽) is located |풓| units from the pole.  If 풓 > ퟎ, the point lies on the terminal side of 휽.  If 풓 < ퟎ, the point lies along the ray opposite the terminal side of 휽.  If 풓 = ퟎ, the point lies at the pole, regardless of the value of 휽.

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Chapter 5 Complex Numbers, Polar Coordinates, and Parametric Equations 5.3 Polar Coordinates Plot the points with the following polar coordinates:

(ퟑ, ퟑퟏퟓ°) (−ퟐ, 흅)

흅 (−ퟏ, − ) ퟐ

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Chapter 5 Complex Numbers, Polar Coordinates, and Parametric Equations 5.3 Polar Coordinates

2 Find multiple sets of polar coordinates for a given point.

(풓, 휽) = (풓, 휽 + ퟐ흅) (풓, 휽) = (−풓, 휽 + 흅) 1 Adding 1 revolution, or 2휋 radians, to the angle Adding revolution, or 휋 radians, to the angle and does not change the point’s location. 2 replacing 푟 with – 푟 does not change the point’s location.

To find two other representations for the point (풓, 휽),  Add ퟐ흅 to the angle and do not change r  Add 흅 to the angle and replace 풓 with – 풓.

Multiple Representation of Points If n is any , the point (풓, 휽) can be represented as (풓, 휽) = (풓, 휽 + ퟐ풏흅) or (풓, 휽) = (−풓, 휽 + 흅 + ퟐ풏흅)

흅 Find another representation of (ퟓ, ) in which ퟒ

a. 푟 is positive and 2휋 < 휃 < 4휋.

b. 푟 is negative and 0 < 휃 < 2휋.

c. 푟 is positive and −2휋 < 휃 < 0.

풙ퟐ + 풚ퟐ = 풓ퟐ 풙 퐜퐨퐬 휽 = 풓 풚 퐬퐢퐧 휽 = 풓 풚 퐭퐚퐧 휽 = 풙

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Chapter 5 Complex Numbers, Polar Coordinates, and Parametric Equations 5.3 Polar Coordinates Relations Between Polar and Rectangular Coordinates 풙 = 풓 퐜퐨퐬 휽 풚 = 풓 퐬퐢퐧 휽 풙ퟐ + 풚ퟐ = 풓ퟐ 풚 퐭퐚퐧 휽 = 풙

3 Convert a point from polar to rectangular coordinates.

Find the rectangular coordinates of the point with the following polar coordinates: (ퟑ, 흅)

흅 Find the rectangular coordinates of the point with the following polar coordinates: (−ퟖ, ) ퟑ

4 Convert a point from rectangular to polar coordinates.

There are infinitely many representations for a point in polar coordinates. If the point (푥, 푦) lies in one of the four quadrants, we will use the representation in which  r is a positive, and  휃 is the smallest positive angle with the terminal side passing through (푥, 푦).

Converting a point from Rectangular to polar Coordinates (풓 > ퟎ 퐚퐧퐝 ퟎ ≤ 휽 ≤ ퟐ흅) 1. Plot the point (푥, 푦).

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Chapter 5 Complex Numbers, Polar Coordinates, and Parametric Equations 5.3 Polar Coordinates 2. Find r by computing the distance from the to (푥, 푦): 푟 = √푥2 + 푦2. 푦 3. Find θ using tan 휃 = With the terminals side of θ passing through (푥, 푦). 푥

Find polar coordinates of the point whose rectangular coordinates are (ퟏ, −√ퟑ).

5 Convert an equation from rectangular to polar coordinates.

A polar equation is an equation whose variables are r and θ. To convert a rectangular equation in x and y to a polar equation in r and θ, replace x with 풓 퐜퐨퐬 휽 and y with 풓 퐬퐢퐧 휽. ퟓ 풓 = 풓 = 퐜퐬퐜 휽 퐜퐨퐬 휽 + 퐬퐢퐧 휽 Convert the following rectangular equation to a polar equation that expresses r in terms of θ. ퟑ풙 − 풚 = ퟔ

6 Convert an equation 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 θ. We use one or more of the following equations: 풚 풓ퟐ = 풙ퟐ + 풚ퟐ 풓 퐜퐨퐬 휽 = 풙 풓 퐬퐢퐧 휽 = 풚 퐭퐚퐧 휽 = . 풙 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.

Convert the polar equation to a rectangular equation in x and y: 풓 = ퟒ 5

Chapter 5 Complex Numbers, Polar Coordinates, and Parametric Equations 5.3 Polar Coordinates

Convert the polar equation to a rectangular equation in x and y: 풓 = −ퟔ 퐜퐨퐬 휽

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