3. In polar coordinates, we label a point P by coordinates (r, θ)
In polar coordinates, we label a point P by coordinates (r, θ). where r is the distance to the origin O and θ is the angle between
and the positive x-axis. By convention, an angle is positive if the corresponding rotation is counterclockwise. We call r the radial coordinate and θ the angular coordinate. cv cv cv cv cv cv

5. By definition,
If r > 0, the angular coordinate θ
of P = (x, y) is
From Rectangular to Polar Coordinates Find the polar coordinates of point P.

6. A few remarks are in order before proceeding:
The angular coordinate is not unique because (r, θ) and (r, θ + 2πn) label the same point for any integer n. For instance, point P has radial coordinate r = 2, but its angular coordinate can be any one of
The origin O has no well-defined angular coordinate, so we assign to O the polar coordinates (0, θ) for any angle θ.
By convention, we allow negative radial coordinates. By definition, (−r, θ) is the reflection of (r, θ) through the origin. With this convention, (−r, θ) and (r, θ + π) represent the same point.
We may specify unique polar coordinates for points other than the origin by placing restrictions on r and θ. We commonly choose r > 0 and 0 ≤ θ < 2π.

7. When determining the angular coordinate of a point P = (x, y), remember that there are two angles between 0 and 2π satisfying tan θ = y/x. You must choose θ so that (r, θ) lies in the quadrant containing P.
Choosing θ Correctly Find two polar representations of P = (−1, 1), one with r > 0 and one with r < 0.

8. A curve is described in polar coordinates by an equation involving r and θ, which we call a polar equation. By convention, we allow solutions with r < 0.
A line through the origin O has the simple equation θ = θ0, where θ0 is the angle between the line and the x-axis. Indeed, the points with θ = θ0 are (r, θ0), where r is arbitrary (positive, negative, or zero).
Line Through the Origin Find the polar equation of the line through the origin of slope

9. To describe lines that do not pass through the origin, we note that any such line has a unique point P0 that is closest to the origin. The next example shows how to write down the polar equation of the line in terms of P0.
Line Not Passing Through O
Show that
is the polar equation of the line
whose
point closest to the origin is

10. Find the polar equation of the line.
is the polar equation of the line
whose
point closest to the origin is

11. Often, it is hard to guess the shape of a graph of a polar equation
Often, it is hard to guess the shape of a graph of a polar equation. In some cases, it’s helpful to rewrite the equation in rectangular coordinates.
Identify the curve with polar equation r = 2a cos θ
(a a constant).
Multiply the equation by r to obtain r2 = 2ar cos θ.

12. Symmetry About the x-Axis Sketch the limaçon curve r = 2 cos θ − 1.
Step 1. Plot points.
Step 2. Analyze r
as a function of θ.
Step 3. Use symmetry.
The curve r = 2 cos θ − 1 is called the limaçon, from the Latin word for “snail.” It was first described in 1525 by the German artist Albrecht Dürer.