Problem 16 - Calculus Early Transcendentals

Question

\(15-20\) Identify the curve by finding a Cartesian equation for the curve. $$ r \cos \theta=1 $$

Step-by-Step Solution

Answer

The curve is a vertical line at \( x = 1 \).

Step 1: Recognize the Polar Equation

The given equation is \( r \cos \theta = 1 \). This equation is in polar form, with \( r \) representing the radial distance from the origin and \( \theta \) representing the angle from the positive x-axis.

Step 2: Use Polar to Rectangular Conversion Formulas

Recall the conversion formulas between polar and Cartesian coordinates: \( x = r \cos \theta \) and \( y = r \sin \theta \). These formulas allow us to translate between polar and Cartesian coordinates.

Step 3: Substitute for x in the Equation

Since \( x = r \cos \theta \), we can substitute \( x \) for \( r \cos \theta \) in the given equation. Thus, the equation becomes \( x = 1 \).

Step 4: Interpret the Cartesian Equation

The Cartesian equation \( x = 1 \) signifies a vertical line in the xy-plane. This line is parallel to the y-axis and intercepts the x-axis at \( x = 1 \). Thus, the curve represented by the polar equation \( r \cos \theta = 1 \) is a vertical line.

Key Concepts

Converting Polar Equations to Cartesian Equations

Converting polar equations into Cartesian equations involves translating the way we express positions and curves in a coordinate system. In polar coordinates, each point is defined by a radius \( r \) and an angle \( \theta \), indicating the distance and direction from the origin. When converting to Cartesian coordinates, we need to adopt a different approach.

To perform this conversion:

• We utilize two key formulas: \( x = r \cos \theta \) and \( y = r \sin \theta \). These relate the polar coordinates to the Cartesian system by expressing them in terms of \( x \) and \( y \).
• Given a polar equation, identify parts of the equation that fit these conversion formulas.
• Substitute these expressions to switch from polar to Cartesian forms, allowing the equation to be understood and plotted in the Cartesian plane.

In the example \( r \cos \theta = 1 \), substituting \( x \) for \( r \cos \theta \) gives us \( x = 1 \). Understanding this translation aids greatly in characterizing curves across different coordinate systems.

Understanding Polar Coordinates

Polar coordinates offer a unique method to define a point or curve based on its distance and angle from the origin. Instead of using perpendicular x and y axes like Cartesian coordinates, polar coordinates describe each position with \( r \) (the radial distance) and \( \theta \) (the angular measure).

Key aspects of polar coordinates include:

• The radial distance \( r \): This is a measure of how far the point is from the origin, similar to how the radius extends outwards from the center of a circle.
• The angle \( \theta \): This specifies the direction relative to the positive x-axis, reflecting which "direction" the point lies in.

Imagine pinpointing a location on a map using distance and direction instead of a typical grid. This often makes polar coordinates especially useful in scenarios involving rotational movement or circular paths.

By appreciating these elements, we can better grasp polar equations and their interpretations as they often represent curves like spirals, circles, and lines intrinsically related to a center point.

Characterizing Curves in Cartesian Plane

Once we have converted a polar equation into a Cartesian equation, characterizing the curve becomes straightforward if you understand basic Cartesian plane geometry. In the Cartesian system, a point is correlated to two orthogonal axes labeled as x and y.

For the equation \( x = 1 \), which translates from \( r \cos \theta = 1 \):

• This reveals that every point on the curve has the x-coordinate equal to 1, meaning it is a vertical line.
• The line intersects the x-axis at \( x = 1 \), showing its location is fully defined along the vertical extent of the y-axis.

Vertical lines, created by an equation like \( x = k \) where \( k \) is a constant, always run parallel to the y-axis and are a basic form of curve in the Cartesian plane.

Understanding this fundamental concept helps in analyzing more complex geometries and in easily plotting or visualizing these curves through Cartesian coordinates.