Problem 23 - Calculus

Question

Problems 23-28 are about polar equations that give circles.] Convert \(r=\sin \theta\) into an \(x y\) equation. Multiply first by \(r\).

Step-by-Step Solution

Answer

The Cartesian equation is \(x^2 + y^2 - y = 0\).

Step 1: Understand the Polar Equation

The polar equation given is \(r = \sin \theta\). This means that the length of the radius \(r\) is equal to the sine of the angle \(\theta\).

Step 2: Multiply by r

To convert the equation into Cartesian coordinates, first multiply both sides by \(r\), giving us \(r^2 = r \sin \theta\).

Step 3: Substitute Cartesian Coordinates

Recall the conversions from polar to Cartesian coordinates: \(x = r \cos \theta\) and \(y = r \sin \theta\). Also, note that \(r^2 = x^2 + y^2\).

Step 4: Replace Polar Terms with Cartesian Terms

Substitute \(r^2 = x^2 + y^2\) and \(r \sin \theta = y\) into the equation \(r^2 = r \sin \theta\). This gives us \(x^2 + y^2 = y\).

Step 5: Rearrange the Equation

Rearrange the equation \(x^2 + y^2 = y\) to \(x^2 + y^2 - y = 0\). This is the equation of the circle in Cartesian coordinates.

Key Concepts

Cartesian Coordinates

Cartesian coordinates are a way to determine every point in a two-dimensional plane using an ordered pair of numbers. These coordinates

• consist of an x-coordinate (horizontal direction)
• and a y-coordinate (vertical direction).

It’s like plotting points on a graph where each point is represented by \((x, y)\).
For example, the point (3, 4) means moving 3 units along the x-axis and 4 units up the y-axis.
This system is pivotal for equations involving circles, lines, and parabolas, enabling easy visualization and translation from abstract equations to tangible plots.

Equation Conversion

Equation conversion involves changing mathematical expressions from one form to another, often to simplify the problem-solving process. When dealing with polar to Cartesian conversions, it's important to understand the relationship between the coordinates.

• The polar coordinate system uses \( r \) (distance from the origin) and \( \theta \) (angle from the positive x-axis).
• Cartesion conversions use the formulas \(x = r \cos \theta \)
• and \( y = r \sin \theta \).

For the problem where we have \(r = \sin \theta\), we aim to convert it into Cartesian form.
By initially multiplying the polar equation by \( r \), the expression becomes \(r^2 = r \sin \theta\).
Through substitution, we use \(r^2 = x^2 + y^2\) and \(r \sin \theta = y\), resulting in \(x^2 + y^2 = y\). Converting equations helps reveal further properties of the graph, such as symmetry and shape.

Circle Equations

Circle equations generally describe a circle in analytic geometry. They define all points on a plane at a specific distance (radius) from a center point. The standard form for a circle's equation is \((x-h)^2 + (y-k)^2 = r^2\), where

• \((h, k)\) is the circle's center,
• and \(r\) is its radius.
• For circles centered at the origin \((0,0)\),
• the equation simplifies to \(x^2 + y^2 = r^2\).

In the current problem, we converted from polar to Cartesian form or \(x^2 + y^2 = y\), neatly fitting the equation of a circle.
The translation of such an equation is crucial for accurate graph plotting and measuring relative distances and areas within geometry problems.