Further Applications of Trigonometry

# Polar Coordinates

### Learning Objectives

In this section, you will:

• Plot points using polar coordinates.
• Convert from polar coordinates to rectangular coordinates.
• Convert from rectangular coordinates to polar coordinates.
• Transform equations between polar and rectangular forms.
• Identify and graph polar equations by converting to rectangular equations.

Over 12 kilometers from port, a sailboat encounters rough weather and is blown off course by a 16-knot wind (see (Figure)). How can the sailor indicate his location to the Coast Guard? In this section, we will investigate a method of representing location that is different from a standard coordinate grid.

### Plotting Points Using Polar Coordinates

When we think about plotting points in the plane, we usually think of rectangular coordinatesin the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems. In this section, we introduce to polar coordinates, which are points labeledand plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane.

The polar grid is scaled as the unit circle with the positive x-axis now viewed as the polar axis and the origin as the pole. The first coordinateis the radius or length of the directed line segment from the pole. The angle measured in radians, indicates the direction ofWe move counterclockwise from the polar axis by an angle ofand measure a directed line segment the length ofin the direction ofEven though we measurefirst and then the polar point is written with the r-coordinate first. For example, to plot the pointwe would moveunits in the counterclockwise direction and then a length of 2 from the pole. This point is plotted on the grid in (Figure).

### Plotting a Point on the Polar Grid

Plot the pointon the polar grid.

The angleis found by sweeping in a counterclockwise direction 90° from the polar axis. The point is located at a length of 3 units from the pole in thedirection, as shown in (Figure).

### Try It

Plot the pointin the polar grid.

### Plotting a Point in the Polar Coordinate System with a Negative Component

Plot the pointon the polar grid.

We know thatis located in the first quadrant. However,We can approach plotting a point with a negativein two ways:

1. Plot the pointby movingin the counterclockwise direction and extending a directed line segment 2 units into the first quadrant. Then retrace the directed line segment back through the pole, and continue 2 units into the third quadrant;
2. Movein the counterclockwise direction, and draw the directed line segment from the pole 2 units in the negative direction, into the third quadrant.

See (Figure)(a). Compare this to the graph of the polar coordinateshown in (Figure)(b).

### Try It

Plot the pointsandon the same polar grid.

### Converting from Polar Coordinates to Rectangular Coordinates

When given a set of polar coordinates, we may need to convert them to rectangular coordinates. To do so, we can recall the relationships that exist among the variablesand

Dropping a perpendicular from the point in the plane to the x-axis forms a right triangle, as illustrated in (Figure). An easy way to remember the equations above is to think ofas the adjacent side over the hypotenuse andas the opposite side over the hypotenuse.

### Converting from Polar Coordinates to Rectangular Coordinates

To convert polar coordinatesto rectangular coordinates let

### How To

Given polar coordinates, convert to rectangular coordinates.

1. Given the polar coordinate writeand
2. Evaluateand
3. Multiplybyto find the x-coordinate of the rectangular form.
4. Multiplybyto find the y-coordinate of the rectangular form.

### Writing Polar Coordinates as Rectangular Coordinates

Write the polar coordinatesas rectangular coordinates.

The rectangular coordinates areSee (Figure).

### Writing Polar Coordinates as Rectangular Coordinates

Write the polar coordinatesas rectangular coordinates.

See (Figure). Writing the polar coordinates as rectangular, we have

The rectangular coordinates are also

### Try It

Write the polar coordinatesas rectangular coordinates.

### Converting from Rectangular Coordinates to Polar Coordinates

To convert rectangular coordinates to polar coordinates, we will use two other familiar relationships. With this conversion, however, we need to be aware that a set of rectangular coordinates will yield more than one polar point.

### Converting from Rectangular Coordinates to Polar Coordinates

Converting from rectangular coordinates to polar coordinates requires the use of one or more of the relationships illustrated in (Figure).

### Writing Rectangular Coordinates as Polar Coordinates

Convert the rectangular coordinatesto polar coordinates.

We see that the original pointis in the first quadrant. To finduse the formulaThis gives

To findwe substitute the values forandinto the formulaWe know thatmust be positive, asis in the first quadrant. Thus

So,andgiving us the polar pointSee (Figure).

#### Analysis

There are other sets of polar coordinates that will be the same as our first solution. For example, the pointsandwill coincide with the original solution ofThe pointindicates a move further counterclockwise bywhich is directly oppositeThe radius is expressed asHowever, the angleis located in the third quadrant and, asis negative, we extend the directed line segment in the opposite direction, into the first quadrant. This is the same point asThe pointis a move further clockwise byfromThe radius,is the same.

### Transforming Equations between Polar and Rectangular Forms

We can now convert coordinates between polar and rectangular form. Converting equations can be more difficult, but it can be beneficial to be able to convert between the two forms. Since there are a number of polar equations that cannot be expressed clearly in Cartesian form, and vice versa, we can use the same procedures we used to convert points between the coordinate systems. We can then use a graphing calculator to graph either the rectangular form or the polar form of the equation.

### How To

Given an equation in polar form, graph it using a graphing calculator.

1. Change the MODE to POL, representing polar form.
2. Press the Y= button to bring up a screen allowing the input of six equations:
3. Enter the polar equation, set equal to
4. Press GRAPH.

### Writing a Cartesian Equation in Polar Form

Write the Cartesian equationin polar form.

The goal is to eliminateandfrom the equation and introduceand Ideally, we would write the equationas a function ofTo obtain the polar form, we will use the relationships betweenandSince andwe can substitute and solve for

Thus,andshould generate the same graph. See (Figure).

To graph a circle in rectangular form, we must first solve for

Note that this is two separate functions, since a circle fails the vertical line test. Therefore, we need to enter the positive and negative square roots into the calculator separately, as two equations in the formandPress GRAPH.[/hidden-answer]

### Rewriting a Cartesian Equation as a Polar Equation

Rewrite the Cartesian equationas a polar equation.

This equation appears similar to the previous example, but it requires different steps to convert the equation.

We can still follow the same procedures we have already learned and make the following substitutions:

Therefore, the equationsand should give us the same graph. See (Figure).

The Cartesian or rectangular equation is plotted on the rectangular grid, and the polar equation is plotted on the polar grid. Clearly, the graphs are identical.[/hidden-answer]

### Rewriting a Cartesian Equation in Polar Form

Rewrite the Cartesian equationas a polar equation.

We will use the relationships and

### Try It

Rewrite the Cartesian equationin polar form.

### Identify and Graph Polar Equations by Converting to Rectangular Equations

We have learned how to convert rectangular coordinates to polar coordinates, and we have seen that the points are indeed the same. We have also transformed polar equations to rectangular equations and vice versa. Now we will demonstrate that their graphs, while drawn on different grids, are identical.

### Graphing a Polar Equation by Converting to a Rectangular Equation

Covert the polar equation to a rectangular equation, and draw its corresponding graph.

The conversion is

Notice that the equationdrawn on the polar grid is clearly the same as the vertical linedrawn on the rectangular grid (see (Figure)). Just asis the standard form for a vertical line in rectangular form,is the standard form for a vertical line in polar form.

A similar discussion would demonstrate that the graph of the function will be the horizontal lineIn fact, is the standard form for a horizontal line in polar form, corresponding to the rectangular form[/hidden-answer]

### Rewriting a Polar Equation in Cartesian Form

Rewrite the polar equationas a Cartesian equation.

The goal is to eliminateandand introduce and We clear the fraction, and then use substitution. In order to replace with and we must use the expression

The Cartesian equation isHowever, to graph it, especially using a graphing calculator or computer program, we want to isolate

When our entire equation has been changed fromandto and we can stop, unless asked to solve foror simplify. See (Figure).

The “hour-glass” shape of the graph is called a hyperbola. Hyperbolas have many interesting geometric features and applications, which we will investigate further in Analytic Geometry.[/hidden-answer]

#### Analysis

In this example, the right side of the equation can be expanded and the equation simplified further, as shown above. However, the equation cannot be written as a single function in Cartesian form. We may wish to write the rectangular equation in the hyperbola’s standard form. To do this, we can start with the initial equation.

### Try It

Rewrite the polar equation in Cartesian form.

or, in the standard form for a circle,

### Rewriting a Polar Equation in Cartesian Form

Rewrite the polar equationin Cartesian form.

This equation can also be written as

Access these online resources for additional instruction and practice with polar coordinates.

### Key Equations

 Conversion formulas

### Key Concepts

• The polar grid is represented as a series of concentric circles radiating out from the pole, or origin.
• To plot a point in the formmove in a counterclockwise direction from the polar axis by an angle of and then extend a directed line segment from the pole the length of in the direction of Ifis negative, move in a clockwise direction, and extend a directed line segment the length of in the direction of See (Figure).
• Ifis negative, extend the directed line segment in the opposite direction ofSee (Figure).
• To convert from polar coordinates to rectangular coordinates, use the formulasandSee (Figure) and (Figure).
• To convert from rectangular coordinates to polar coordinates, use one or more of the formulas:andSee (Figure).
• Transforming equations between polar and rectangular forms means making the appropriate substitutions based on the available formulas, together with algebraic manipulations. See (Figure), (Figure), and (Figure).
• Using the appropriate substitutions makes it possible to rewrite a polar equation as a rectangular equation, and then graph it in the rectangular plane. See (Figure), (Figure), and (Figure).

### Section Exercises

#### Verbal

How are polar coordinates different from rectangular coordinates?

For polar coordinates, the point in the plane depends on the angle from the positive x-axis and distance from the origin, while in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there are infinite representations.

How are the polar axes different from the x– and y-axes of the Cartesian plane?

Explain how polar coordinates are graphed.

Determinefor the point, then moveunits from the pole to plot the point. Ifis negative, moveunits from the pole in the opposite direction but along the same angle. The point is a distance ofaway from the origin at an angle offrom the polar axis.

How are the pointsandrelated?

Explain why the pointsandare the same.

The pointhas a positive angle but a negative radius and is plotted by moving to an angle ofand then moving 3 units in the negative direction. This places the point 3 units down the negative y-axis. The pointhas a negative angle and a positive radius and is plotted by first moving to an angle ofand then moving 3 units down, which is the positive direction for a negative angle. The point is also 3 units down the negative y-axis.

#### Algebraic

For the following exercises, convert the given polar coordinates to Cartesian coordinates withandRemember to consider the quadrant in which the given point is located when determiningfor the point.

For the following exercises, convert the given Cartesian coordinates to polar coordinates withRemember to consider the quadrant in which the given point is located.

For the following exercises, convert the given Cartesian equation to a polar equation.

For the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.

orcircle

line

line

hyperbola

circle

line

#### Graphical

For the following exercises, find the polar coordinates of the point.

For the following exercises, plot the points.

For the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.

For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.

#### Technology

Use a graphing calculator to find the rectangular coordinates ofRound to the nearest thousandth.

Use a graphing calculator to find the rectangular coordinates ofRound to the nearest thousandth.

Use a graphing calculator to find the polar coordinates ofin degrees. Round to the nearest thousandth.

Use a graphing calculator to find the polar coordinates ofin degrees. Round to the nearest hundredth.

Use a graphing calculator to find the polar coordinates ofin radians. Round to the nearest hundredth.

#### Extensions

Describe the graph of

Describe the graph of

A vertical line withunits left of the y-axis.

Describe the graph of

Describe the graph of

A horizontal line withunits below the x-axis.

What polar equations will give an oblique line?

For the following exercise, graph the polar inequality.

### Glossary

polar axis
on the polar grid, the equivalent of the positive x-axis on the rectangular grid
polar coordinates
on the polar grid, the coordinates of a point labeledwhereindicates the angle of rotation from the polar axis andrepresents the radius, or the distance of the point from the pole in the direction of
pole
the origin of the polar grid