Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ x=-3 $$

Step-by-Step Solution

Answer

The polar equation is \(r\cos(\theta) = -3\).

Step 1: Understand the Cartesian equation

The given Cartesian equation is a vertical line at \(x = -3\). This means that any point on this line has its x-coordinate fixed at -3.

Step 2: Convert Cartesian Coordinates to Polar Coordinates

In polar coordinates, any point can be represented as \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle from the positive x-axis. The conversion formulae from Cartesian coordinates \((x, y)\) to polar coordinates are: \[x = r\cos(\theta)\]\[y = r\sin(\theta)\]This will help us convert the line equation \(x = -3\) to polar form.

Step 3: Use the Conversion Formula

Since \(x = -3\), replace \(x\) with the polar equivalent:\[r\cos(\theta) = -3\]This is our polar equation.

Step 4: Simplify the Polar Equation

The equation \(r\cos(\theta) = -3\) is already in its simplest form as a polar equation. It describes a vertical line at all points where \(r\cos(\theta) = -3\), equivalent to \(x = -3\) in Cartesian coordinates.

Key Concepts

Cartesian to Polar conversion

The conversion from Cartesian to polar coordinates involves expressing an equation from the familiar

• x-axis
• y-axis

system into another system where points are described by their radial distance and angle from the origin. In mathematics, Cartesian coordinates are often expressed as a pair

• \((x, y)\)

while polar coordinates use

• \((r, \theta)\)

. In this system,

• \(r\)

is the radial distance from the origin, and

• \(\theta\)

is the angle measured from the positive x-axis. This conversion is pivotal when equations, like lines or curves, are easier to analyze or visualize in a certain coordinate system.
Understanding the conversion formulas is crucial in this process. The equations

• \(x = r\cos(\theta)\)
• \(y = r\sin(\theta)\)

give us a direct method to translate Cartesian coordinates to polar coordinates. For example, when a line in Cartesian form is

• \(x = -3\)

,we substitute

• \(x\)

with

• \(r\cos(\theta)\)

to obtain the polar equation

• \(r\cos(\theta) = -3\)

. This represents the same geometrical shape, just in the polar system.

vertical lines in polar coordinates

Vertical lines, in the context of the polar coordinate system, have a unique representation. In the Cartesian coordinate system, vertical lines are defined by equations like

• \(x = c\)

where

• \(c\)

is a constant. Every location along a vertical line has a fixed x-coordinate. However, in polar coordinates, we translate this into an expression involving both the radial coordinate

• \(r\)

and the angle

• \(\theta\)

.
For example, consider the line given by

• \(x = -3\)

. In polar coordinates, this is expressed as

• \(r\cos(\theta) = -3\)

. This equation tells us that the line consists of all the points

• \((r, \theta)\)

where, when calculating the projection along the x-axis using

• \(\cos(\theta)\)

, the result is always

• \(-3\)

. This elegant expression highlights how concepts like vertical lines extend into the polar realm.

polar coordinate system

The polar coordinate system is a mathematical framework used to describe the location of a point in a plane using a distance and an angle. Unlike the Cartesian system that uses perpendicular axes, the polar system uses:

• \(r\) representing the radial distance from the origin, often called the pole.
• \(\theta\), which denotes the angle formed with the positive x-axis.

This system is particularly useful when dealing with problems or equations involving circles, spirals, or any features with rotational symmetry.
When engaging with polar coordinates, one often translates between the polar and Cartesian systems by employing formulas such as:

• \(x = r\cos(\theta)\)
• \(y = r\sin(\theta)\)

These expressions allow one to map a point specified by

• \((r, \theta)\)

back to

• \((x, y)\)

in the Cartesian system, thus enabling a more complete understanding of spatial relationships.
Overall, whether working with simple lines or more complex figures, the polar coordinate system provides a versatile lens for interpreting and visualizing different geometrical forms.