Question

The letters \(x\) and \(y\) represent rectangular coordinates. Write each equation using polar coordinates \((r, \theta) .\) $$ x^{2}+y^{2}=x $$

Short answer

The equation in polar coordinates is \( r = \cos(\theta) \).

Step-by-step solution

Convert Cartesian Coordinates to Polar Coordinates

Recall the polar coordinate conversion formulas: - For the radial coordinate: \( r = \sqrt{x^2 + y^2} \) - For the angular coordinate: \( \theta = \tan^{-1}(\frac{y}{x}) \) Also, note the relationship between Cartesian and polar coordinates: \( x = r \cos(\theta) \) \( y = r \sin(\theta) \)

Substitute Polar Coordinates into the Given Equation

The given equation is: \( x^2 + y^2 = x \) Substitute \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \): \((r \cos(\theta))^2 + (r \sin(\theta))^2 = r \cos(\theta)\)

Simplify the Equation

Simplify the equation using the Pythagorean identity: \( r^2 \cos^2(\theta) + r^2 \sin^2(\theta) = r \cos(\theta) \) Since \( \cos^2(\theta) + \sin^2(\theta) = 1 \): \( r^2 = r \cos(\theta) \)

Solve for r

Divide both sides of the equation by \( r \) (assuming \( r e 0 \)): \( r = \cos(\theta) \)

Key concepts

rectangular coordinates

Rectangular coordinates, also known as Cartesian coordinates, are used to define the location of a point in a plane using two perpendicular axes. These axes are usually labeled as the x-axis (horizontal) and the y-axis (vertical). The coordinates of a point are written as \( (x, y) \), where x represents the distance along the horizontal axis and y represents the distance along the vertical axis.
Here are some key points to remember:

• Rectangular coordinates are easy to understand and visualize.
• They are commonly used in algebra and geometry.
• Each pair of coordinates corresponds to a single point in the plane.

The given exercise initially uses the equation \(x^2 + y^2 = x\) which is expressed in rectangular coordinates.

Cartesian coordinates

The terms Cartesian coordinates and rectangular coordinates can be used interchangeably. They both describe a system for representing points in a plane using two intersecting lines, known as axes. These coordinates were popularized by the French mathematician René Descartes, hence the name Cartesian.
The Cartesian coordinate system allows us to graph equations and visualize their solutions. Here are some important aspects:

• The origin (0,0) is the point where the x-axis and y-axis intersect.
• Positive x-values are to the right of the origin, while negative x-values are to the left.
• Positive y-values are above the origin, while negative y-values are below.

In this context, you can transition from Cartesian coordinates to polar coordinates to solve certain types of equations more easily.

polar coordinate conversion formulas

Polar coordinates offer a different way to describe points in a plane using a radius and angle. To convert Cartesian coordinates \( (x, y) \) to polar coordinates \( (r, \theta) \), you can use the following formulas:

• \( r = \sqrt{x^2 + y^2} \) (radius)
• \( \theta = \tan^{-1}(\frac{y}{x}) \) (angle)

Additionally, to express x and y in terms of polar coordinates, use:

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

For the given problem, the received equation is expressed as: \( x^2 + y^2 = x \). By substituting \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \), we simplify the equation using trigonometric identities.

simplifying equations

Simplifying equations helps make them easier to understand and solve. Here are some steps:

• Identify substitutions that can simplify the terms in the equation.
• Use trigonometric identities to combine like terms.
• Isolate the variable you are solving for.

In our exercise, after substituting polar coordinates into the equation \( x^2 + y^2 = x \), we get:
\[ (r \cos(\theta))^2 + (r \sin(\theta))^2 = r \cos(\theta) \]

Using the Pythagorean identity \( \cos^2(\theta) + \sin^2(\theta) = 1 \), the equation becomes:
\[ r^2 = r \cos(\theta) \]

By dividing both sides by \( r \) (assuming \( r ≠ 0 \)), we find:
\[ r = \cos(\theta) \]

This result is much simpler and shows the relationship between r and \( \theta \).