Problem 19 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-26: Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$ x+y=3 $$

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{3}{\cos \theta + \sin \theta} \).

Step 1: Recall conversion formulas

To convert the given cartesian equation to polar coordinates, recall the formulas: \( x = r \cos \theta \) and \( y = r \sin \theta \).

Step 2: Substitute polar formulas

Replace \( x \) with \( r \cos \theta \) and \( y \) with \( r \sin \theta \) in the equation. The cartesian equation \( x + y = 3 \) becomes \( r \cos \theta + r \sin \theta = 3 \).

Step 3: Factor out \( r \)

Factor \( r \) from the terms on the left side of the equation: \( r (\cos \theta + \sin \theta) = 3 \).

Step 4: Solve for \( r \)

Solve the equation for \( r \) by dividing both sides by \( \cos \theta + \sin \theta \): \( r = \frac{3}{\cos \theta + \sin \theta} \).

Step 5: Present the polar equation

The polar equation that represents the same line as the cartesian equation is \( r = \frac{3}{\cos \theta + \sin \theta} \).

Key Concepts

Cartesian Coordinates

The Cartesian coordinate system is commonly used in mathematics to define the position of points in a plane. It is characterized by two intersecting perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). Every point in this plane is represented by a pair of numbers

• x-coordinate: Specifies the point's position along the horizontal axis.
• y-coordinate: Specifies the point's position along the vertical axis.

Let's consider the equation given in the problem:
\(x + y = 3\). This equation describes a straight line in the Cartesian plane where, for any point lying on this line, the sum of its x-coordinate and y-coordinate equals 3. Cartesian coordinates are essential in determining tangible distances and angles using basic algebra, which makes them extremely useful for visualizing and solving geometric problems.

Coordinate Conversion

Coordinate conversion is a crucial concept that involves transitioning from one coordinate system to another. In this exercise, we are converting from Cartesian coordinates to polar coordinates, which capture the position of points using distance and angle.
The conversion formulas we'll use are:

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

Here, \(r\) represents the radial distance from the origin to the point, and \(\theta\) represents the angle between the positive x-axis and the line connecting the origin to the point. This conversion is useful in simplifying complex mathematical problems, as polar coordinates can often make calculus and analysis more intuitive. Following the conversion steps in the solution helps transform the equation \(x + y = 3\) from Cartesian to its equivalent polar form, allowing different yet equivalent visualizations of the line.

Polar Equations

Polar equations are utilized to describe curves using polar coordinates. These equations express the relationship between the radial distance \(r\) and the angle \(\theta\) in the polar plane.
In this exercise, the task was to find a polar equation equivalent to the Cartesian equation \(x + y = 3\). Through conversion and simplifications, we derived the polar equation: \[r = \frac{3}{\cos \theta + \sin \theta}\]
This equation plot will result in a line equivalent to the original line in the Cartesian system. Polar equations are particularly useful in dealing with scenarios that involve circular symmetry or periodic phenomena. They help simplify complex systems by focusing on angular and radial aspects rather than horizontal and vertical positions, thus enabling the solving of advanced geometrical and physical problems.