Pre-Calc Exam Notes 146

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

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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. ...
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