(X, Y), We Describe a Point by (R, Θ) in the Polar Coordinates Where R Is Its Dis- Tance from the Origin and Θ Is the Angle It Makes with the Positive X−Axis

Lecture 32: Polar Coordinates (11.3)

Instead of using (x, y), we describe a point by (r, θ) in the polar coordinates where r is its dis- tance from the origin and θ is the angle it makes with the positive x−axis.

The Cartesian Coordinates of a point (x, y) and its polar coordinates (r, θ) are related by the equations

x = r cos θ , y = r sin θ. y r2 = x2 + y2 , tan θ = , x =6 0. x

1 Unlike in the rectangular coordinates where each point (x, y) is uniquely deﬁned, a point in polar coordinates can have inﬁnitely many description. For example, (r, θ) = (1, 0) = (1, 2nπ) . In fact, (r, θ) = (r, θ ± 2nπ) = (−r, θ ± (2n + 1)π)

Negative r : By convention, r > 0, ”−r” means swing to the other side of angle θ.(−r, θ) is the reﬂection of (r, θ) through the origin. (−r, θ) and (r, θ + π) represent the same point.

The pole(the origin) is denoted by (0, θ) for any angle θ. ex. Plot the points whose polar coordinates are (r, θ) = (1, 0), (1, 2π), (1, −4π), (0, π), (0, π/2), (0, 7.4π), π π π (−1, − 2 ), (−1, 2 ), (4, π)(−4, 0), (2, − 3 )

2 ex. Find 2 other pairs of polar coordinates of the point (−1, −π/2), one with r > 0, and one with r < 0.

2π ex. Convert the point (r, θ) = (3, 3 ) from polar coordinates to Cartesian Coordinates.

3 √ ex. Convert the point (x, y) = (−1, − 3) from Cartesian Coordinates to polar coordinates.

ex. Find a polar equation for the curve repre- sented by the given Cartesian equation. x2 + y2 = 4

4 ex. Graph the polar curves by ﬁnding a Carte- sian equation for the curve: 1. r = 2

2. r = 3 cos θ

5 ex. Construct a polar equation for a circle of radius a with its center at x = a, y = 0.

Circle family : r = a, r = 2a sin θ, r = 2a cos θ. r = a is a circle centered at the origin. Others are ’oﬀ-centered circles. The choice of cos θ or sin θ determines if the circle sits on the x− or y−axis.

6 Symmetry in Polar graphs: 1. (r, θ) and (r, −θ) are symmetric with respect to the x−axis.

2. (r, θ) and (−r, θ) are symmetric with respect to the pole.

3. (r, θ) and (r, π − θ) are symmetric to the y− axis.

ex. Graph the polar curve r = cos 2θ.

Method 1: Plot points: Since cosine is an even function, cos 2θ = cos 2(−θ); (r, θ) and (r, −θ) are both on the curve. The curve is symmetric w.r.t. the x−axis. Using the symmetry, it suﬃces to plot points for 0 ≤ θ ≤ π.

7 Method 2: Analyze r as a function of θ in rectangular coordinates. ( to be continued ···)