Problem 24 - Precalculus: Functions and Graphs

Question

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$2 y=-x+4$$

Step-by-Step Solution

Answer

The polar equation is \(r = \frac{4}{\cos\theta + 2\sin\theta}\).

Step 1: Convert the Cartesian Equation

Start by rewriting the given Cartesian equation in standard form. The original equation is \(2y = -x + 4\). To express this in terms of both \(x\) and \(y\), rewrite it as \(x + 2y = 4\).

Step 2: Recall Polar-Cartesian Relations

In polar coordinates, we use the relations: \(x = r\cos\theta\) and \(y = r\sin\theta\). These relate the coordinates \((x, y)\) in Cartesian to \(r\) and \(\theta\) in polar coordinates.

Step 3: Substitute Polar Expressions

Substitute the expressions for \(x\) and \(y\) from Step 2 into the Cartesian equation \(x + 2y = 4\). This becomes: \(r\cos\theta + 2r\sin\theta = 4\).

Step 4: Factor Out and Solve for r

Factor out \(r\) from the polar equation: \(r(\cos\theta + 2\sin\theta) = 4\). Next, solve for \(r\) to get: \(r = \frac{4}{\cos\theta + 2\sin\theta}\). This is the required polar equation.

Key Concepts

Cartesian to polar conversion

When converting Cartesian coordinates to polar coordinates, it's crucial to understand how these two systems relate to each other. The Cartesian coordinate system uses the familiar

• horizontal axis, "x"
• vertical axis, "y"

In contrast, the polar coordinate system defines points based on their distance from the origin and the angle from the positive x-axis.
Cartesians can be transformed into polar coordinates using the following equations:

• The distance from the origin, denoted as "r", is calculated using: \( r = \sqrt{x^2 + y^2} \).
• The angle from the positive x-axis, denoted as "\( \theta \)", is calculated as: \( \theta = \tan^{-1}(\frac{y}{x}) \).

By understanding these conversions, you can easily translate equations from the x-y plane to the r-\( \theta \) plane.
This process allows you to express the relations more conveniently when dealing with circular or radial problems.

Polar coordinates

Polar coordinates provide a valuable alternative to Cartesian coordinates, especially when dealing with curves centered around a point, such as circles and spirals. In polar coordinates:

• "r" measures the radial distance from the origin to the point.
• "\( \theta \)" specifies the direction angle measured counter-clockwise from the positive x-axis.

For example, when given a Cartesian equation, transforming it into polar form helps to visualize shapes that are inherently circular or symmetrical around a central point.
This transformation simplifies the mathematics, making it easier to analyze and understand the geometry of these curves. Hence, polar coordinates are not just a different way to plot points but a powerful tool to simplify complex mathematical problems.

Trigonometric identities

Trigonometric identities play a pivotal role in the conversion process between coordinate systems. These identities allow for the simplification of expressions involving angles and radii when dealing with polar and Cartesian systems. Some of the basic trigonometric identities frequently used include:

• \( \sin^2\theta + \cos^2\theta = 1 \)
• \( 1 + \tan^2\theta = \sec^2\theta \)
• \( \sin2\theta = 2\sin\theta\cos\theta \)

These identities help to convert and manipulate equations, making math involving angles more manageable.
For the polar equation conversion, specifically,

• "x" is expressed as \( r\cos\theta \)
• "y" as \( r\sin\theta \)

Recognizing and applying these trigonometric identities is essential, as seen in the conversion where the equation \( x + 2y = 4 \) becomes \( r(\cos\theta + 2\sin\theta) = 4 \).
Thus, trigonometric identities serve as a mathematical bridge that aids in translating complex Cartesian equations into simpler polar forms.