Polar Coordinates and Conversions in Precalculus

Chapter 6: Additional Topics in Trigonometry

Section 6.3: Polar Coordinates

This section introduces the polar coordinate system, methods for plotting points, and conversions between polar and rectangular coordinates. It also covers how to convert equations between these coordinate systems.

Objectives

• Plot points in the polar coordinate system.

• Find multiple sets of polar coordinates for a given point.

• Convert a point from polar to rectangular coordinates.

• Convert a point from rectangular to polar coordinates.

• Convert an equation from rectangular to polar coordinates.

• Convert an equation from polar to rectangular coordinates.

Polar Coordinate System

Definition and Structure

The polar coordinate system is based on a fixed point called the pole (analogous to the origin in rectangular coordinates) and a fixed ray called the polar axis (usually the positive x-axis). A point P in the plane is represented by an ordered pair (r, θ), where:

• r: The directed distance from the pole to the point P.

• θ: The angle measured from the polar axis to the line segment connecting the pole to P. Angles can be measured in degrees or radians.

• Positive angles are measured counterclockwise from the polar axis; negative angles are measured clockwise.

Plotting Points in the Polar Coordinate System

To plot a point with polar coordinates (r, θ):

• Draw the angle θ from the polar axis.

• Move a distance |r| from the pole along the terminal side of the angle.

• If r is negative, move in the direction opposite the terminal side of θ.

Example 1a: Plotting (3, 315°)

• Draw θ = 315° counterclockwise from the polar axis.

• Since r = 3 is positive, plot the point three units out along the terminal side of 315°.

Example 1b: Plotting (-2, π)

• Draw θ = π (180°) counterclockwise from the polar axis.

• Since r = -2 is negative, plot the point two units in the direction opposite the terminal side of π.

Example 1c: Plotting (-1, -π/2)

• Draw θ = -π/2 (clockwise 90° from the polar axis).

• Since r = -1 is negative, plot the point one unit in the direction opposite the terminal side of -π/2.

Multiple Representations of Polar Coordinates

Any point in the plane can be represented by infinitely many polar coordinates. This is due to the periodic nature of angles and the sign of r.

• Adding or subtracting multiples of $2 ext{π}$ (or 360°) to θ gives the same direction.

• Changing the sign of r and adding π (or 180°) to θ also locates the same point.

Example 2a: Find another representation of $\left(5, \frac{\pi}{4}\right)$ with $r > 0$ and $2\pi < \theta < 4\pi$

• Add $2\pi$ to θ: $\left(5, \frac{\pi}{4} + 2\pi\right) = \left(5, \frac{9\pi}{4}\right)$

Conversions Between Polar and Rectangular Coordinates

Formulas

• From polar to rectangular:

$x = r \cos \theta$

$y = r \sin \theta$

• From rectangular to polar:

$r = \sqrt{x^2 + y^2}$

$\theta = \arctan\left(\frac{y}{x}\right)$

Example: Polar to Rectangular Conversion

• Given (3, π):

$x = 3 \cos \pi = 3(-1) = -3$

$y = 3 \sin \pi = 3(0) = 0$

Rectangular coordinates: (-3, 0)

Example: Rectangular to Polar Conversion

• Given (1, -√3):

$r = \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = 2$

$\theta = \arctan\left(\frac{-\sqrt{3}}{1}\right) = -\frac{\pi}{3}$

Polar coordinates: (2, $-\frac{\pi}{3}$)

Converting Equations Between Coordinate Systems

Rectangular to Polar

To convert a rectangular equation in x and y to polar form:

• Replace $x$ with $r \cos \theta$

• Replace $y$ with $r \sin \theta$

Example: Convert $3x - y = 6$ to polar form

$3r \cos \theta - r \sin \theta = 6$

$r(3 \cos \theta - \sin \theta) = 6$

$r = \frac{6}{3 \cos \theta - \sin \theta}$

Polar to Rectangular

To convert a polar equation to rectangular form:

• Replace $r$ with $\sqrt{x^2 + y^2}$

• Replace $\cos \theta$ with $\frac{x}{r}$

• Replace $\sin \theta$ with $\frac{y}{r}$

• Replace $\tan \theta$ with $\frac{y}{x}$

Example: Convert $r = 10 \sin \theta$ to rectangular form

$r = 10 \sin \theta$

Substitute $r$ and $\sin \theta$:

$\sqrt{x^2 + y^2} = 10 \frac{y}{\sqrt{x^2 + y^2}}$

Multiply both sides by $\sqrt{x^2 + y^2}$:

$x^2 + y^2 = 10y$

Summary Table: Polar and Rectangular Conversions

Rectangular	Polar
$x = r \cos \theta$	$r = \sqrt{x^2 + y^2}$
$y = r \sin \theta$	$\theta = \arctan\left(\frac{y}{x}\right)$
$x^2 + y^2 = r^2$	$\cos \theta = \frac{x}{r}$, $\sin \theta = \frac{y}{r}$

Additional info: The notes are based on textbook slides and cover all major aspects of polar coordinates relevant to a Precalculus course, including definitions, plotting, and conversions. Examples are expanded for clarity and completeness.