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Chapter 10: Problem 18

Find the Cartesian equations of the graphs of the given polar equations. $$ \text { 3. } r=3 $$

Short Answer

Expert verified
The Cartesian equation is \( x^2 + y^2 = 9 \).

Step by step solution

01

Understand the Polar Equation

The given polar equation is \( r = 3 \). This equation represents a circle in the polar coordinate system with a radius of 3, centered at the pole (origin).
02

Recall Conversion Formulas

To convert from polar to Cartesian coordinates, use the formulas: \( x = r \cos \theta \) and \( y = r \sin \theta \). Also, note that \( r = \sqrt{x^2 + y^2} \).
03

Substitute and Simplify

Substitute the expression for \( r \) from the equation \( r = 3 \) into the equation for \( r \) in Cartesian form: \[ \sqrt{x^2 + y^2} = 3 \].
04

Remove the Square Root and Simplify

Square both sides to remove the square root: \( x^2 + y^2 = 9 \). This is the equation of a circle with radius 3 centered at the origin in the Cartesian plane.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polar Coordinates
Polar coordinates are a way to represent points in a plane using a distance and an angle. Imagine you're plotting a point on a circle. Instead of using just a grid (like in Cartesian coordinates), you use the angle from a fixed direction and the radius (distance from a center point, usually the origin). This makes polar coordinates really useful for circular and spiral patterns.

In polar coordinates, a point is defined as either \((r, \theta)\) or \((r, \phi)\), where:
  • \(r\) is the radial distance from the origin.
  • \(\theta\) or \(\phi\) is the angle measured from the positive x-axis (counter-clockwise).
For example, the polar equation \(r = 3\) represents all points that are 3 units away from the origin at any angle. This traces out a circle because, regardless of the angle, the distance remains the same.
Conversion Formulas
Converting polar coordinates to Cartesian coordinates involves translating the radial and angular information into x and y values on the Cartesian plane. The formulas for these conversions are:
  • \(x = r \cos \theta\)
  • \(y = r \sin \theta\)
These formulas help map each point defined in polar form to its corresponding point in the Cartesian system. You can also revert back to polar from Cartesian using:
  • \(r = \sqrt{x^2 + y^2}\)
  • \(\theta = \tan^{-1}(\frac{y}{x})\)
Using the example from the exercise, the polar equation \(r = 3\) converts by applying the formula \(r = \sqrt{x^2 + y^2}\) to confirm that any point with this radial distance forms a circle in Cartesian coordinates. After substituting and simplifying, you derive the equation \(x^2 + y^2 = 9\).
Circle Equation
In Cartesian coordinates, a circle's equation is typically written as \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle, and \(r\) is the radius. If the circle is centered at the origin \((0,0)\), the equation simplifies to \(x^2 + y^2 = r^2\).

For instance, the solution \(x^2 + y^2 = 9\) represents a circle in the Cartesian plane with:
  • Radius \(r = 3\)
  • Center at the origin \((0,0)\)
This shows how a simple polar equation like \(r = 3\) can directly translate to a familiar Cartesian form. Both forms are efficient in their own contexts鈥攚hether you're dealing with circular shapes or linear algebraic expressions.

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Most popular questions from this chapter

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