Problem 18 - Precalculus: Functions and Graphs

Question

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$x^{2}+y^{2}=2$$

Step-by-Step Solution

Answer

The polar equation is \(r = \sqrt{2}\).

Step 1: Understand the Given Cartesian Equation

The given equation is a circle equation in its standard form: \(x^2 + y^2 = r^2\) where \(r\) is the radius of the circle. In this case, \(r^2 = 2\), so the radius \(r\) is \(\sqrt{2}\).

Step 2: Convert Cartesian Coordinates to Polar Coordinates

In polar coordinates, the variables \(x\) and \(y\) are expressed in terms of \(r\) (radius) and \(\theta\) (angle) as: \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\).

Step 3: Substitute Polar Coordinates into the Cartesian Equation

Substitute \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\) into the equation \(x^2 + y^2 = 2\):\[ (r\cos(\theta))^2 + (r\sin(\theta))^2 = 2 \]

Step 4: Simplify the Equation

Simplify \((r\cos(\theta))^2 + (r\sin(\theta))^2 = 2\) to find:\[ r^2(\cos^2(\theta) + \sin^2(\theta)) = 2 \]Since \(\cos^2(\theta) + \sin^2(\theta) = 1\), this simplifies further to:\[ r^2 = 2 \]

Step 5: Solve for the Polar Equation

Since \(r^2 = 2\), the polar equation that corresponds to the given Cartesian equation is:\[ r = \sqrt{2} \]

Key Concepts

Conversion of Coordinates

In mathematics, converting coordinates from one system to another can help simplify problems or provide better insights. Two common systems are Cartesian and polar coordinates.

**Cartesian coordinates** are based on a grid, where a point is defined by an \(x, y\) pair representing horizontal and vertical distances from an origin. Most students are familiar with this from graphing in geometry.

**Polar coordinates**, on the other hand, define a point by its distance from the origin and the angle this line makes with the positive x-axis. This system is especially useful for circular or rotational contexts.

When converting from Cartesian to polar, we use the following relationships:

• \( x = r\cos(\theta) \)
• \( y = r\sin(\theta) \)
• The radius \( r \) can be determined by \( r = \sqrt{x^2 + y^2} \)
• The angle \( \theta \) can be found using \( \theta = \tan^{-1}(\frac{y}{x}) \)

These conversions allow us to take any point given in one system and express it in the other, facilitating the solving of equations like circles, ellipses, and other shapes that are more naturally described in one system or the other.

Circle Equation

A circle's equation describes all the points on the circumference that are equidistant from a central point. In Cartesian coordinates, the standard form of a circle's equation is \(x^2 + y^2 = r^2\), where \(r\) is the radius.

This form makes it easy to see and understand how the circle is structured:

• Every point \( (x, y) \) on the circle's edge is the same distance \( r \) from the center point, which is usually the origin \( (0, 0) \)
• Knowing \(r^2\) allows us to quickly sketch or visualize the circle's size and shape

The task of converting a Cartesian circle equation into a polar equation involves recognizing that \(x^2 + y^2\) can be directly replaced with \(r^2\) in polar coordinates. When we look at the original circle equation \(x^2 + y^2 = 2\):

• The given \(r^2 = 2\) simply implies that in polar form, the radius is \(r = \sqrt{2}\)

This transformation allows us to express circular formations and equations in a format that remains consistent with their inherent characteristics in polar coordinates.

Cartesian Coordinates

Cartesian coordinates form the backbone of many mathematical plotting systems. Named after René Descartes, this system uses a straight-line grid to define points in a plane by two numbers, \(x\) and \(y\).

The axes dividing the plane into four quadrants illustrate how points are plotted:

• The horizontal axis (\