Polar Coordinates - Chapter 8 Chapter 8 - Complex Numbers and Polar Graphs

Learning Objectives

1. Introduce the polar coordinate system and plot points.

2. Convert between rectangular coordinates and polar coordinates.

3. Find equivalent equations using polar coordinates.

In mathematics, position in two-dimensional space is usually expressed using rectangular (Cartesian) coordinates. However, the rectangular coordinate system is not the only way to identify points in the plane. In this section we define a new positional system that can be used to easily represent relations that are not functions.

Objective: Introduce the polar coordinate system and plot points.

The polar coordinate system is a positional system based on a point, called the pole, and a ray called the polar axis. Given any other point in a plane, another ray can be constructed from the pole through that given point. Then the point can be described using coordinates that consist of a directed distance r from the pole to the point along the ray, and angle θ from the polar axis to the ray. That is, points are expressed as ordered pairs in the form

( )

^r^,^θ ^.

The polar axis is drawn to coincide with the positive x-axis where any point in the plane can be used to form an angle in standard position. Furthermore, it is important to note that the polar coordinate system allows for r to be negative. When the directed distance r is negative, the terminating ray is extended 180° in the opposite direction and the point is located r units from the pole in this opposite direction.

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In the same way rectangular graph paper aids in graphing with rectangular coordinates, polar graph paper aids in graphing using polar coordinates and consists of concentric circles and lines at the following special angles. [Link to polar graph paper.]

Use the above construction to graph points expressed with polar coordinates.

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Example: Plot the following points using the polar coordinate system.

(

5, 45°

)

(

⁻^{4, 60}^°

)

^c.

(

^{3, 135}⁻ ^°

)

Solution:

a. Here r=5 units and θ =45°.

b. Since r is negative, mark the point 4 units in the opposite direction as the terminal ray indicated by θ =60°.

c. In this case, mark a point 3 units along the terminal side of θ = −135° drawn in standard position.

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It is not uncommon for the angle in polar coordinates to be expressed using radian measure.

Try this! Plot 5,⁵ 6

− π 

 

  Answer:

Points expressed using the rectangular coordinate system are unique, that is, there is only one way to express any given point

( )

x y . This is not the case with the polar coordinate system; ^, points can be expressed in many different ways. In other words, polar coordinates are not unique.

Example: Give three equivalent points using polar coordinates:

(

^{4, 75° .}

)

Solution: A conterminal angle in standard position can be calculated by subtracting 360°.

75° −360° = −285° Therefore, the point

(

^{4, 285}⁻ ° locates the same point.

)

Certainly, we could obtain many more coordinate pairs locating the same point simply by adding and subtracting multiples of 360° to 75°. However, we can also express this point using

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a negative value for r where the angle terminates on the opposite ray that lies along the same line. To do this, add and subtract 180° to 75° and user= − . 4

75 180 255 75 180 105

⁻^5,135^{° ,}

) (

^{− −}^{5, 225}^°

)

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Objective: Convert between polar coordinates and rectangular coordinates.

Any point in a plane can be expressed using polar coordinates or rectangular coordinates. To convert between systems, we use the relationships given by the general definition of the trigonometric functions.

Recall that,

cos x

θ = r sin y

θ = r Multiplying both sides by r we have,

cos

x=r θ y=rsinθ

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These equations can be used to convert from polar coordinates to rectangular coordinates.

Example: Convert the given polar coordinates to rectangular coordinates.

(

Solution: After applying the square root property to r² =x²+ we have, y²

2 2

r= ± x +y

This allows us to determine the directed distance r using a positive or negative value. However, for simplicity, we will choose r to be positive using r= x² +y² and determine the angle θ

( ) ^{( )}

tan 5 undefined 0

Objective: Find equivalent equations using polar coordinates.

In algebra, we study equations that involve the variables x and y, where y is often expressed as a function of x:

(

)

² ⁵

y= x− + y= − x+3 y=log2

( )

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Such equations are called rectangular equations and are graphed in the rectangular coordinate plane. Equations that are expressed using polar coordinates r and θ are called polar equations.

Typically, polar equations express r in terms of θ:

5sin

r= θ r= −2 3cosθ r=cos 5θ

In general, polar equations are useful when graphing relations that are not functions in rectangular coordinates. Consider the following example of a simple polar equation

r=θ

Graphing this equation using the polar coordinate system produces the Archimedean spiral below.

Pictured: r =θ (radians) [Archimedean Spiral Interactive]

This curve fails the vertical line test for functions in rectangular coordinates, and therefore cannot be expressed as^y⁼ ^{f x}

( )

Our goal is to produce sketches of many curves using polar coordinates. To do this we will need to convert equations from rectangular form to polar form. To convert an equation in rectangular form we use the facts,

cos

x=r θ and y=rsinθ Substitute these equations in and then solve for r.

Example: Express the equation of the line 2x−3y=1 in polar form.

Solution: Substitute x=rcosθ and y=rsinθ, and then solve for r.

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( ) ( )

Next we consider circles centered at the origin.

Example: Express the equation of the circle x²+y² =25 in polar form.

Note that the equation of a circle centered at the origin is simple in polar coordinates as compared to rectangular coordinates.

Answer: r=5

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Try this! Write the unit circle x²+y² = in polar form. 1 Answer: r= 1

In general, any circle centered at the origin will have the form x²+y² = . Converting this to a² polar form we have,

( ) ( )

As a final note, we point out that the shape of a graph in the plane is the same using polar coordinates or rectangular coordinates. In other words, it does not matter which positional system is used to produce the sketch.

4 (Rectangular) 4 (Polar)

25 (Rectangular) 5 (Polar)

Key Takeaways ***** ***** *****

• Points in the plane expressed using polar coordinates consist of a directed distance r and an angle θ. These points are written as

( )

^r^,^θ and are not uniquely identified. determine the correct angle θ after finding the reference angle.

• The shape of a graph in the plane is the same using polar coordinates or rectangular coordinates.

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Section Exercises Part A

Plot the set of points using the polar coordinate system.

(

^{3, 60° ,}

) (

^{2,150° ,}

) (

^{5, 90°}

)

)

²⁺^y² ⁼⁹

62. ^x²⁺

(

^y⁻²

)

² ⁼⁴

Express the given equation in rectangular form.

63. r=7 64. r= 2

65. 4

cos 3sin

r= θ θ

−

66. ⁷

2 cos sin

r⁼ θ − θ

67. ⁵

1 sin r⁼ + θ

68. ²

1 sin r⁼ − θ 69. r=6 cosθ 70. r=2 sinθ 71. r=4 sinθ 72. r =10 cosθ 73. ^r² ⁼^{6 sin 2}

( )

^θ

74. ^r² ⁼^{2 cos 2}

( )

^θ

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Answers to Section Exercises

Part C

In document Chapter 8 Chapter 8 - Complex Numbers and Polar Graphs (Page 47-63)