POLAR COORDINATES Dr. Shildneck

The Polar Coordinate System 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. A point P in the polar coordinate system is represented by the ordered pair (r, Ѳ). r is a directed distance from the pole. Ѳ is the angle from the polar axis to the line segment that passes from the pole to P. The angle can be measured in either degrees or radians. Positive angles are measured counterclockwise from the polar axis. Negative angles are measured clockwise from the polar axis. (r, Ѳ) are the polar coordinates of P. r Ѳ Pole Polar Axis

Plotting Points in A Polar Coordinate System As a result of measuring distances and then rotating them, the polar coordinate system is made up of a set of concentric circles, set equal distances apart from one another, with the pole at their centers. The distance between circles is (typically) one unit. Typical polar graph papers include directional lines every 15o (or π/2 radians). The horizontal and vertical lines through the pole are the same axes as the x and y axes on rectangular systems.

Plotting Points in A Polar Coordinate System r can be positive or negative. |r| = distance from the pole. If r is positive, the point is IN the direction of the terminal side. If r is negative, the point is in the OPPOSITE direction as the terminal side. Ѳ is the angle of rotation. When plotting the point (r, Ѳ), it is often easier to move in the opposite order. Starting at the pole, rotate and look in the direction of Ѳ. Apply r… If r is positive, move that direction r steps. If r is negative, move backwards r steps.

Plotting Points Examples (3, 45o) (-5, 5π/6) Start at the pole… Rotate to the angle… Go the right number of steps…

Coordinates have multiple representations Since r can be positive or negative, and Since Ѳ can rotate either direction... Every point has infinitely different representations. By changing the angle Ѳ by 360 degrees, we can use the same r-value with a different angle. By changing the angle Ѳ by 180 degrees, we can use the opposite r-value with a different angle. If n is any integer, the point (r, Ѳ) can be represented as (r, Ѳ) = (r, Ѳ + 2πn) or (r, Ѳ) = (-r, Ѳ + π + 2πn)

Coordinates have multiple representations Find 3 different coordinates that represent the position shown on the graph.

Polar and Rectangular Coordinates The polar coordinate system represents the same locations that our Cartesian coordinate system does. The only difference is how the position is indicated. The origin in the rectangular system is the same as the pole in the polar. Since the rectangular system is represented by horizontal (x) and vertical (y) movements based on the origin, and the polar system is represented by angles and “slanted” movements, there are algebraic/trigonometric conversion formulas that describe the relationship between the two systems.

Polar and REctangular Some coordinates are easy to transform between systems. Points on either the x or y axes are simple. To change other points, you must know the relationships.

Polar and REctangular For non-axis points, you must consider the relationships between the angle, the distance out, horizontal distance, x, and vertical distance, y.

Polar and Rectangular Coordinates The relationship between the three distances (x, y, and r) is the Pythagorean theorem. The relationship between the angle (Ѳ) and the x and y is the tangent.

Polar and Rectangular Coordinates Convert the following polar coordinates to rectangular. Convert the following rectangular coordinates to polar (nearest tenth degree).

Polar Equations A Polar Equation is an equation whose variables are r and theta. There are a number of different types of polar equations, but most are written with r in terms of theta. Examples of polar equations include…

Polar and Rectangular Equations It is often easier to draw the graph of an equation in terms of polar coordinates rather than rectangular (and vice versa). For this purpose, we need to be able to convert equations between systems. Sometimes we can use the conversions for polar coordinates to help do this. But, most of the time, we need more “handy” formulas to transform equations. These equations include our “typical” formulas for x and y (that we saw when working with vectors) in terms of the sine and cosine.

Rectangular to Polar conversions Example 1

Rectangular to Polar conversions Example 2

Polar to Rectangular Conversions Example 1

Polar to Rectangular Conversions Example 1

Polar to Rectangular Conversions Example 1

Polar to Rectangular Conversions Example 1

Assignment 1 Handout Textbook Page #3, 9, 19, 21 ,23, 25, 31, 41, 47, 51, 53, 55, 61, 71, 73, 75, 77-80 all