Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the Cartesian equations of the graphs of the given polar equations. $$ r \cos \theta+3=0 $$

Step-by-Step Solution

Answer

The Cartesian equation is \(x + 3 = 0\).

Step 1: Understand Polar Equation

The given polar equation is \( r \cos \theta + 3 = 0 \). Our aim is to convert this polar equation to its Cartesian form. Remember that the polar coordinates \((r, \theta)\) relate to the Cartesian coordinates \((x, y)\) through the following relationships: \(x = r \cos \theta\) and \(y = r \sin \theta\).

Step 2: Identify Relation in Polar

The term \(r \cos \theta\) in the polar equation corresponds to \(x\) in Cartesian coordinates. Therefore, replace \(r \cos \theta\) with \(x\).

Step 3: Formulate Cartesian Equation

Substituting \(x\) for \(r \cos \theta\), we rewrite the equation as \(x + 3 = 0\). This is now in a standard Cartesian form.

Key Concepts

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system. They represent points in a plane by an angle and a radius. In this system, a point is described as \((r, \theta)\). Here, \(r\) represents the distance from the origin (or pole), and \(\theta\) represents the angle made with the positive x-axis.
This coordinate system is especially useful for dealing with problems involving circles and periodic patterns.

• \(r\) can be positive, negative, or zero.
• \(\theta\) is usually measured in radians but can also be in degrees.

Polar coordinates simplify the process of working with curves that are circular or spiral in nature.

Conversion to Cartesian Coordinates

Converting from polar coordinates to Cartesian coordinates is a common task in mathematics. This transformation helps us to reframe problems into a more familiar rectilinear (straight-line) perspective.
The conversion formulas are derived from basic trigonometry:

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

By substituting the polar terms \(r \cos \theta\) with \(x\) and \(r \sin \theta\) with \(y\), you can easily translate between polar and Cartesian equations.
This conversion is crucial when analyzing or graphing equations that originate in a polar format.

Polar Equations

Polar equations express relationships between \(r\) and \(\theta\). They often describe conic sections like circles, ellipses, and hyperbolas.
For example, an equation like \(r = 2 + 3 \cos \theta\) describes a limaçon, a type of spiral.

• Polar equations take different forms depending on the problem context.
• Writing polar coefficients allows for operational symmetry in solving problems linked to rotation or spirals.

Understanding how these equations relate back to Cartesian space helps leverage both coordinate systems effectively.

Cartesian Coordinates

Cartesian coordinates describe a point as \((x, y)\) in a two-dimensional grid. This system, developed by René Descartes, provides a straightforward way to express geometric locations.

• \(x\) represents the horizontal position.
• \(y\) indicates the vertical position.

In Cartesian coordinates, graphs represent relationships by plotting points along perpendicular x and y axes. Understanding this system is foundational for geometry and algebra, crucial when solving equations derived from other coordinate systems.

Mathematics Education

Learning about different coordinate systems enhances spatial reasoning and problem-solving skills. By bridging polar and Cartesian coordinates, students gain insight into how different mathematical perspectives offer unique solutions.
This conversion process exemplifies algebraic manipulation and fosters analytical abilities that are pivotal in higher mathematics.

• Engaging with both systems deepens understanding of functions and graphs.
• Visualizing changes between coordinate perspectives augments comprehension.

Mathematics education emphasizes these skills to prepare students for advanced studies in calculus, physics, and engineering.