Problem 23 - Thomas Calculus

Question

Replace the polar equations in Exercises \(23-48\) by equivalent Cartesian equations. Then describe or identify the graph. $$ r \cos \theta=2 $$

Step-by-Step Solution

Answer

The Cartesian equation is \( x = 2 \), representing a vertical line.

Step 1: Convert to Cartesian Coordinates

The given polar equation is \( r \cos \theta = 2 \). Recall that in polar coordinates, \( x = r \cos \theta \). Therefore, substitute \( x = r \cos \theta \) into the equation to get \( x = 2 \).

Step 2: Graph Interpretation

The equation \( x = 2 \) is a vertical line in the Cartesian coordinate system crossing the x-axis at \( x = 2 \). This line extends vertically, parallel to the y-axis.

Key Concepts

Polar Coordinates

Polar coordinates are a way of describing a point in a plane using distance and angle. With a polar coordinate \(r, \theta\), \(r\) represents the distance from the origin, and \(\theta\) the angle from the positive x-axis. This system is useful when dealing with problems involving curves and circles, as it can simplify how we describe circular and rotational motion.

• \( r \) is always a non-negative number.
• Angle \( \theta \) is usually measured in radians or degrees.
• The polar grid has concentric circles where each ring represents a constant value of \( r \).

As we use polar coordinates, we learn that some points can be expressed in multiple ways. For example, the same point can be reached by adding or subtracting full rotations (2π in radians) to the angle θ.

Cartesian Coordinates

Cartesian coordinates describe any point in a plane using two perpendicular axes, usually labeled x and y. This system, based on a grid of vertical and horizontal lines, is most commonly used in algebra and geometry to graph equations.

• \( x \) represents the horizontal distance from the origin.
• \( y \) represents the vertical distance from the origin.
• The intersection of the axes is called the origin, represented by \( (0, 0) \).

Converting an equation from polar to Cartesian involves replacing polar terms with their equivalent Cartesian terms. For instance, in the exercise, we used the relation \( x = r \cos \theta \) to transform the polar equation into a Cartesian one. This conversion is essential in simplifying equations for tasks in calculus or analytic geometry.

Graph Interpretation

Interpreting the graphs of equations is key to understanding geometric relationships in mathematics. Once we have a Cartesian equation, we can determine the shape and position of its graph by examining the equation's structure.
In the case of \( x = 2 \), the graph is a straightforward vertical line that maintains a constant x-coordinate of 2 for every possible y-value. Graphing this in a two-dimensional plane will show:

• The line is parallel to the y-axis.
• It passes through all points where the x-coordinate is 2.
• Each point on this line doesn't depend on a specific y-value, which can be any real number.

Graph interpretation helps visualize algebraic expressions and understand their real-world applications.

Vertical Line in Cartesian Plane

A vertical line in the Cartesian plane is described by an equation like \( x = a \), where \( a \) is a constant. This line features several unique properties:

• The x-coordinate is constant for all points on the line.
• It creates a perpendicular intersection with the x-axis at point \( (a, 0) \).
• Vertical lines have an undefined slope, meaning they rise vertically without running horizontally.

These lines are important because they represent constant values of x, which can often appear in physics or engineering problems where horizontal constraints are considered. Understanding vertical lines also bolsters comprehension of how linear equations function in a broader geometric context.