Problem 59 - A Graphical Approach to Precalculus with Limits

Question

For each equation, find an equivalent equation in rectangular coordinates. Then graph the result. $$r=2 \sin \theta$$

Step-by-Step Solution

Answer

The equivalent equation is \( x^2 + (y-1)^2 = 1 \) and it represents a circle with center at \( (0,1) \) and radius 1.

Step 1: Convert Polar to Rectangular Coordinates

In polar coordinates, we have the relationships: \[ x = r \cos \theta \] and \[ y = r \sin \theta \]Given the polar equation: \[ r = 2 \sin \theta \]Multiply both sides by \( r \) to get: \[ r^2 = 2r \sin \theta \]Now substitute \( r^2 = x^2 + y^2 \) and \( r \sin \theta = y \) to obtain:\[ x^2 + y^2 = 2y \]

Step 2: Rearrange the Equation

Start with the equation: \[ x^2 + y^2 = 2y \]Rearrange it by moving all terms to one side:\[ x^2 + y^2 - 2y = 0 \]

Step 3: Complete the Square

To complete the square for the term \( y^2 - 2y \), take half the coefficient of \( y \) which is \( -1 \), square it to get \( 1\), and add it inside the equation:\[ x^2 + (y^2 - 2y + 1) = 1 \]This results in:\[ x^2 + (y - 1)^2 = 1 \]

Step 4: Identify the Graph

The equation \( x^2 + (y - 1)^2 = 1 \) represents a circle with center at \( (0, 1) \) and radius \( 1 \). Therefore, graph this circle accordingly.

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a way of representing points on a plane using coordinates denoted as \( (x, y) \). These coordinates describe the location of a point based on its horizontal (\(x\)) and vertical (\(y\)) distances from a reference point known as the origin. This coordinate system is also known as the Cartesian coordinate system, named after the French mathematician René Descartes.

The conversion from polar coordinates, which are based on a radius and angle, to rectangular coordinates involves trigonometric functions. Specifically, the relationships used are:

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

Understanding these conversions is crucial in mathematics and physics, where different coordinate systems can offer distinct benefits for solving problems.

Completing the Square

Completing the square is a method used in algebra to transform a quadratic equation into a perfect square trinomial. This technique is particularly useful when working with circle equations, as it simplifies the expression and reveals the circle's geometric properties.

In the case given, we start with the equation \(x^2 + y^2 - 2y = 0\). To complete the square for the \(y\) terms, you take half of the coefficient of \(y\) (which is \(-2\)), divide it by 2 to get \(-1\), and square it, resulting in \(1\). You then add and subtract this square inside the equation:

• \(x^2 + (y^2 - 2y + 1) = 1\)

This transformation results in \((y - 1)^2\), turning the equation into \(x^2 + (y - 1)^2 = 1\). Completing the square helps to rewrite quadratic equations in a form that clearly shows key characteristics such as the center and radius of a circle.

Circle Equation

A circle's equation in rectangular coordinates is typically expressed as \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius. When working with equations derived from polar coordinates, such as our example, this format allows easy identification of a circle's geometric properties.

The given equation, \(x^2 + (y - 1)^2 = 1\), is already in standard circle form. From this, you can immediately observe:

• The center of the circle is at \((0, 1)\).
• The radius is \(1\), since \(r^2 = 1\).

Understanding this form helps in graphing and analyzing circular shapes in various mathematical contexts.

Graphing Polar Equations

Graphing polar equations involves plotting points that are expressed in terms of the radius \(r\) and angle \(\theta\). The given problem involves a polar equation \(r = 2 \sin \theta\), which must be converted to rectangular coordinates for easier graphing. This conversion results in an equation \(x^2 + (y - 1)^2 = 1\), representing a circle in the Cartesian plane.

Understanding how to move between polar and rectangular formats allows one to visualize equations more simply. In polar plots, you interpret the circle by considering how the radius changes with angle. In rectangular coordinates, geometric properties like radius and center are clearly displayed, easing the graphing process.

• This visualization process strengthens comprehension of geometry and trigonometry.
• Mastery of these conversions opens up deeper appreciation of different ways to represent equations.

Graphing polar equations using rectangular coordinates contributes to solving complex mathematical challenges efficiently.