Chap10_Sec3

# Chap10_Sec3 - 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES...

This preview shows pages 1–23. Sign up to view the full content.

PARAMETRIC EQUATIONS PARAMETRIC EQUATIONS AND POLAR COORDINATES AND POLAR COORDINATES 10

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
A coordinate system represents a point in the plane by an ordered pair of numbers called coordinates. PARAMETRIC EQUATIONS & POLAR COORDINATES
Usually, we use Cartesian coordinates, which are directed distances from two perpendicular axes. PARAMETRIC EQUATIONS & POLAR COORDINATES

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Here, we describe a coordinate system introduced by Newton, called the polar coordinate system. It is more convenient for many purposes. PARAMETRIC EQUATIONS & POLAR COORDINATES
10.3 Polar Coordinates In this section, we will learn: How to represent points in polar coordinates. PARAMETRIC EQUATIONS & POLAR COORDINATES

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
POLE We choose a point in the plane that is called the pole (or origin) and is labeled O.
POLAR AXIS Then, we draw a ray (half-line) starting at O called the polar axis. This axis is usually drawn horizontally to the right corresponding to the positive x -axis in Cartesian coordinates.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
ANOTHER POINT If P is any other point in the plane, let: r be the distance from O to P. θ be the angle (usually measured in radians) between the polar axis and the line OP.
POLAR COORDINATES P is represented by the ordered pair ( r , θ ). r , θ are called polar coordinates of P .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
POLAR COORDINATES We use the convention that an angle is: Positive—if measured in the counterclockwise direction from the polar axis. Negative—if measured in the clockwise direction from the polar axis.
If P = O, then r = 0, and we agree that (0, θ ) represents the pole for any value of θ . POLAR COORDINATES

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
We extend the meaning of polar coordinates ( r , θ ) to the case in which r is negative—as follows. POLAR COORDINATES
POLAR COORDINATES We agree that, as shown, the points (– r , θ ) and ( r , θ ) lie on the same line through O and at the same distance | r | from O , but on opposite sides of O .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
POLAR COORDINATES If r > 0, the point ( r, θ ) lies in the same quadrant as θ. If r < 0, it lies in the quadrant on the opposite side of the pole. Notice that ( r, θ ) represents the same point as ( r, θ + π ).
POLAR COORDINATES Plot the points whose polar coordinates are given. a. (1, 5 π /4) b. (2, 3 π ) c. (2, –2 π /3) d. (–3, 3 π /4) Example 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
POLAR COORDINATES The point (1, 5 π /4) is plotted here. Example 1 a
The point (2, 3 π ) is plotted. Example 1 b POLAR COORDINATES

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
POLAR COORDINATES The point (2, –2 π /3) is plotted. Example 1 c
POLAR COORDINATES The point (–3, 3 π /4) is plotted. It is is located three units from the pole in the fourth quadrant. This is because the angle 3 π /4 is in the second quadrant and r = -3 is negative. Example 1 d

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
CARTESIAN VS. POLAR COORDINATES In the Cartesian coordinate system, every point has only one representation. However, in the polar coordinate system, each point has many representations.
CARTESIAN VS. POLAR COORDINATES For instance, the point (1, 5 π /4) in Example 1 a could be written as: (1, –3 π /4), (1, 13 π /4), or (–1, π /4).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
CARTESIAN & POLAR COORDINATES In fact, as a complete counterclockwise rotation is given by an angle 2 π , the point
This is the end of the preview. Sign up to access the rest of the document.