Polar Coordinates

Butterflies are among the most celebrated of all insects. Their symmetry can be explored with trigonometric functions and a system for plotting points called the polar coordinate system.

Objective: Plot each point given in polar coordinates.

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. r can be positive, negative, or zero. Positive angles are measured counterclockwise from the polar axis. Negative angles are measured clockwise from the polar axis.

Plot the points with the following polar coordinates:

Objective: Polar coordinates of a point are given. Find the rectangular coordinates of each point.

If P is a point with polar coordinates, the rectangular coordinates of P are given by

Find the rectangular coordinates of the point with the following polar coordinates:

Objective: The rectangular coordinates of a point are given. Find polar coordinates for each point.

1.To find r, compute the distance from the origin to (x, y). 2.To find, first determine the quadrant that the point lies in.

Find polar coordinates of a point whose rectangular coordinates are:

Find the polar form of the rectangular point (4, 3). Find the polar form for the point (-2, 3). Find the polar coordinates for the rectangular coordinates (-3, 3).

Objective: The letters x and y represent rectangular coordinates. Write each equation using polar coordinates.

Transform the equation from rectangular coordinates to polar coordinates.

Transform the following equations from rectangular coordinates to polar coordinates.

Objective: The letters represent polar coordinates. Write each equation using rectangular coordinates.

Review from Chapter 4 Completing the Square

Transform the equation from polar coordinates to rectangular coordinates.

Transform the following equations from polar coordinates to rectangular coordinates.

Polar Equations and Graphs

An equation whose variables are polar coordinates is called a polar equation. The graph of a polar equation consists of all points whose polar coordinates satisfy the equation.

#1 Identify and graph the equation:

#2 Identify and graph the equation:

#3 Identify and graph the equation:

Graphing a Polar Equation using your Calculator 1.Solve the equation for r in terms of. 2.Clear the memory on your calculator. 3.Select Mode - Polar. 4.Window ( step determines the number of points that your calculator will graph) 5.Zoom - Standard 6.Enter the expression that you found in step 1.

#4 Use your calculator to graph the polar equation:

#5 Use your calculator to graph the polar equation:

Let a be a nonzero real number. Then the graph of the equation is a horizontal line a units above the pole if a > 0 and units below the pole if a < 0. The graph of the equation is a vertical line a units to the right of the pole if a > 0 and units to the left of the pole if a < 0.

#6 Identify and graph the equation:

#7 Identify and graph the equation:

Specific Types of Polar Graphs Symmetry Plot each point given in polar coordinates and find other polar coordinates of the point.