# Further Applications of Trigonometry

## Polar Coordinates

Over 12 kilometers from port, a sailboat encounters rough weather and is blown off course by a 16-knot wind (see ). 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 coordinates 
  in 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 labeled 
  and 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 coordinate 
  is the radius or length of the directed line segment from the pole. The angle 
   measured in radians, indicates the direction of 
  We move counterclockwise from the polar axis by an angle of 
  and measure a directed line segment the length of 
  in the direction of 
  Even though we measure 
  first and then 
   the polar point is written with the r-coordinate first. For example, to plot the point 
  we would move 
  units in the counterclockwise direction and then a length of 2 from the pole. This point is plotted on the grid in .

### 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 variables 
  and 
  
Dropping a perpendicular from the point in the plane to the x-axis forms a right triangle, as illustrated in . An easy way to remember the equations above is to think of 
  as the adjacent side over the hypotenuse and 
  as the opposite side over the hypotenuse.

### 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.

### 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.

###  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.

### Key Equations


### Key Concepts

1. The polar grid is represented as a series of concentric circles radiating out from the pole, or origin.
2. To plot a point in the form 
  move 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 
  If 
  is negative, move in a clockwise direction, and extend a directed line segment the length of 
   in the direction of 
   See .
3. If 
  is negative, extend the directed line segment in the opposite direction of 
  See .
4. To convert from polar coordinates to rectangular coordinates, use the formulas 
  and 
  See  and .
5. To convert from rectangular coordinates to polar coordinates, use one or more of the formulas: 
  and 
  See .
6. Transforming equations between polar and rectangular forms means making the appropriate substitutions based on the available formulas, together with algebraic manipulations. See , , and .
7. 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 , , and .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, convert the given polar coordinates to Cartesian coordinates. Remember to consider the quadrant in which the given point is located when determining 
  for the point.
For the following exercises, convert the given Cartesian coordinates to polar coordinates with 
  Remember 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.

### 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


### Extensions

For the following exercise, graph the polar inequality.