Pre-Calc Exam Notes 146

# Pre-Calc Exam Notes 146 - 146 Chapter 6 • Additional...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 146 Chapter 6 • Additional Topics §6.4 6.4 Polar Coordinates y Suppose that from the point (1, 0) in the x y-coordinate plane we draw a spiral around the origin, such that the distance between any two points separated by 360◦ along the ← 1→ spiral is always 1, as in Figure 6.4.1. We ← 1→ can not express this spiral as y = f ( x) for some function f in Cartesian coordinates, since its graph violates the vertical rule. x 0 1 2 3 However, this spiral would be simple to describe using the polar coordinate system. Recall that any point P distinct from the origin (denoted by O ) in the x y-coordinate plane is a distance r > 0 from the origin, −→ − and the ray OP makes an angle θ with the positive x-axis, as in Figure 6.4.2. We call the pair ( r , θ ) the polar coordinates of P , Figure 6.4.1 and the positive x-axis is called the polar axis of this coordinate system. Note that ( r , θ ) = ( r , θ + 360◦ k) for k = 0, ± 1, ± 2, ..., so (unlike for Cartesian coordinates) the polar coordinates of a point are not unique. y y P (r, θ) r θ O θ x O x −r P (− r , θ ) Figure 6.4.2 Polar coordinates ( r , θ ) Figure 6.4.3 Negative r : (− r , θ ) In polar coordinates we adopt the convention that r can be negative, by deﬁning (− r , θ ) = −→ − ( r , θ + 180◦ ) for any angle θ . That is, the ray OP is drawn in the opposite direction from the angle θ , as in Figure 6.4.3. When r = 0, the point ( r , θ ) = (0, θ ) is the origin O , regardless of the value of θ . You may be familiar with graphing paper, for plotting points or functions given in Cartesian coordinates (sometimes also called rectangular coordinates). Such paper consists of a rectangular grid. Similar graphing paper exists for plotting points and functions in polar coordinates, similar to Figure 6.4.4. ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online