Problem 53 - Algebra and Trigonometry

Question

Convert each rectangular equation to a polar equation that expresses r in terms of \(\theta\). $$ x^{2}+y^{2}=9 $$

Step-by-Step Solution

Answer

The polar equation is \(r = 3\). This corresponds to a circle of radius 3 in the polar coordinate system.

Step 1: Rewrite the Equation using the polar coordinates substitution

Firstly, we need to convert the rectangular equation \(x^2 + y^2 = 9\) to polar coordinates. We know by the Pythagorean theorem, that \(x^2 + y^2 = r^2\). Substituting \(r^2\) into the equation, rewrite the equation as \(r^2 = 9\).

Step 2: Solve for \(r\)

In polar coordinates, it is typical to solve the equation for \(r\). To find \(r\), we take the square root of both sides, \(r = \sqrt{9}\).

Step 3: Finalize the Answer

After taking the square root, we find \(r = 3\) and \(r = -3\). However, since \(r\) cannot be negative, we discard the result \(r = -3\). Therefore, the polar equation that expresses \(r\) in terms of \(\theta\) is \(r = 3\).

Key Concepts

Rectangular Equation

A rectangular equation is an equation composed using Cartesian coordinates. In this system, points are identified using horizontal and vertical axes, known as the x-axis and y-axis. Rectangular equations describe relationships between these x and y coordinates.

For instance, in the given problem, the equation \(x^2 + y^2 = 9\) is a rectangular equation. This equation actually represents a circle centered at the origin of the coordinate system with a radius of 3.

• "x" represents the horizontal distance from the origin.
• "y" represents the vertical distance from the origin.

To solve or analyze such equations in a different mathematical system, one needs to convert them, which we'll discuss next.

Pythagorean Theorem

The Pythagorean theorem is fundamental in mathematics, especially in geometry, and crucial for converting rectangular equations to polar forms. This theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Mathematically, it's written as: \[ a^2 + b^2 = c^2 \] where \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse.

This same principle is adapted when transitioning from Cartesian to polar coordinates. In polar coordinates, the equation \(x^2 + y^2 = r^2\) shows the relationship between the radius \(r\) and the point's horizontal and vertical components \(x\) and \(y\). Hence, using this, we can substitute \(x^2 + y^2\) with \(r^2\) when converting equations.

Converting Equations

Converting equations between rectangular and polar forms is essential in mathematics to visualize different representations of the same pattern or relationship.

When converting a rectangular equation like \(x^2 + y^2 = 9\) to polar coordinates, we substitute polar coordinate equivalents. Use substitution as in:

• \(x = r\cos(\theta)\)
• \(y = r\sin(\theta)\)
• \(x^2 + y^2 = r^2\)

After substituting \(x^2 + y^2 = r^2\), the rectangular equation becomes \(r^2 = 9\). Solving further, we find \(r = \sqrt{9}\) or \(r = 3\).
Notice how replacing these elements allows us to maintain the equation's geometric meaning but in another format.

Coordinate System

A coordinate system is a method of uniquely identifying each point on a plane using numerical coordinates. The two systems commonly used are Cartesian and Polar.

1. **Cartesian Coordinate System**: Each point is defined by a pair of numerical coordinates: \((x, y)\). It represents linear dimensions in straight lines.2. **Polar Coordinate System**: A point is defined by \((r, \theta)\), where \(r\) is the distance from the origin, and \(\theta\) is the angle from the positive x-axis. This is useful for circular or radial patterns.

Understanding both systems helps in translating equations and visualizing graphs differently. In our example, the equation \(x^2 + y^2 = 9\) translates to \(r = 3\), a simple circle in polar form indicating a constant radius of 3 regardless of \(\theta\).
Thus, exploring conversions between these systems can provide insights into the geometry and behavior of mathematical models.