- Topics At a Glance
- Functions
- Increasing or Decreasing or...
- Bounded
- Even and Odd Functions
- Vectors: A New Kind of Animal
- Magnitude
- Direction
- Scaling Vectors
- Unit Vectors
- Vector Notation
- (More than 2)-Dimensional Vectors
- Vector Functions
- Sketching Vector Functions
- Parametric Equations
- Graphing Parametric Equations
- Points on Graphs of Parametric Equations
- Parametrizations of the Unit Circle
- Parameterization of Lines
**Polar Coordinates**- Simple Polar Inequalities
- Switching Coordinates
**Translating Equations and Inequalities between Coordinate Systems**- Polar Functions
- Graphing Polar Functions
- Rules of Graphing We Do (or Don't) Have
- Bounds on Theta
- Intersections of Polar Functions
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem

We can look at rectangular and polar coordinates as two different languages. We use them to describe the same things using different words. In the last section we learned how to translate points from one coordinate system to another.

Begin with a basic example, the right triangle. Rectangular and polar coordinates give different information about a right triangle.

From looking at the triangle we can see that these statements are true:

These **transformation equations** let us translate equations and inequalities between different coordinates.

To translate an equation or inequality from rectangular to polar coordinates, x becomes rcosθ and y becomes rsinθ. We can also replace x^{2} + y^{2} with r^{2}.

Translating from polar to rectangular coordinates isn't quite as straightforward as going the other way. We can replace r^{2} with x^{2} + y^{2}, rcosθ with *x*, and rsinθ with *y*.
However, sometimes we might need to do an extra step or two before we have any recognizable terms to replace.

Why can't we use one coordinate system and be done with it?

The answer is that some equations and inequalities look better in one system than the other. We know from practice that the equation

x^{2} + y^{2} = 36

describes a pizza with a 6 inch radius, but isn't

r = 6

a much nicer way to describe it?

Going the other way,

the polar inequality

r ≤ 2/cosθ

is a mess. If we translate into rectangular coordinates, we find

rcosθ ≤ 2

x≤ &2

which describes the part of the (*x*,*y*)-plane that lies on and to the left of the line *x* = 2:

If we're good at translating between coordinate systems, we can quickly find the simplest representation of a particular equation.

Example 1

Translate the equation |

Example 2

Translate the inequality |

Example 3

Translate the equation rcosθ + rsinθ = 2 into rectangular coordinates. |

Example 4

Translate the inequality into rectangular coordinates, assuming . |

Exercise 1

- Translate each equation into polar coordinates.

- (
*x*+ 2)^{2}+ (*y*+ 2)^{2}= 4 *x*+*y*^{2}= 0- (
*x*+*y*)^{2}= 4

Exercise 2

- Translate each inequality into polar coordinates. %Graph the region being described by the inequality.

*y*≥ 5*x*+*y*≤ 3*x*^{2}+*y*^{2}< 100

Exercise 3

- Translate each equation into rectangular coordinates.

- rcosθ + r
^{2}sin^{2}θ = 4rsinθ - (hint: multiply through by
*r*) - r = 4 (hint: square both sides.)

Exercise 4

- Translate each inequality into rectangular coordinates. Assume
*r*is non-negative.%Graph the region being described by the inequality.

- sin θ < 1/r
- rcos
^{2}θ + rsin^{2}θ = 1 - 5r
^{2}cos^{2}θ + 10rcosθ + 5 -4r^{2}sin^{2}θ + 16rsinθ-16 = 20