Problem 27 - Thomas Calculus

Question

Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian equations. Then describe or identify the graph. $$r \cos \theta=2$$

Step-by-Step Solution

Answer

The polar equation \( r \cos \theta = 2 \) is equivalent to the Cartesian equation \( x = 2 \), a vertical line.

Step 1: Identify the Polar Equation

The given polar equation is \( r \cos \theta = 2 \). In polar coordinates, \( r \) is the radius, and \( \theta \) is the angle.

Step 2: Use the Polar to Cartesian Conversion Formulas

In order to convert to Cartesian coordinates, use the conversion formulas: \( x = r \cos \theta \) and \( y = r \sin \theta \).

Step 3: Substitute and Simplify

Since \( r \cos \theta = x \), substitute \( x \) into the equation: \( x = 2 \). This is the Cartesian equation.

Step 4: Identify the Graph

The equation \( x = 2 \) describes a vertical line in the Cartesian plane, specifically a line parallel to the y-axis that intercepts the x-axis at \( x = 2 \).

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing points on a plane by specifying the distance from a reference point and the angle from a reference direction. This system uses two values:

• \( r \) - The distance from the origin, often referred to as the radius.
• \( \theta \) - The angle measured from the positive x-axis, generally in radians.

The origin in polar coordinates is called the pole and corresponds to the point (0,0) in Cartesian coordinates. Polar coordinates are especially useful in situations where relationships are radial, such as in circular movements or when dealing with waves.
For instance, a point in polar coordinates is given as \( (r, \theta) \). If \( r = 3 \) and \( \theta = \frac{\pi}{4} \), this point would be 3 units away from the origin, at an angle of \( \frac{\pi}{4} \) radians from the positive x-axis. Converting between polar and Cartesian coordinates often involves trigonometric functions and can help reveal different properties of the graph when switched.

Cartesian Coordinates

Cartesian coordinates describe a point in a plane using two perpendicular axes, usually labeled as the x-axis and y-axis. Each point is defined by an ordered pair \( (x, y) \), where:

• \( x \) represents the horizontal position, positive to the right, and negative to the left.
• \( y \) represents the vertical position, positive upwards, and negative downwards.

The Cartesian system is named after René Descartes and is useful for algebraic manipulation and calculations, particularly in engineering and physics.
For example, in the Cartesian coordinate system, the equation \( x = 2 \) specifies a vertical line crossing the x-axis at the point \( (2, 0) \). This powerful system of coordinates allows us to work with a myriad of equations and solutions in various fields, enabling precise graph identification and transformations like the one we have performed from polar equations.

Graph Identification

Graph identification involves recognizing and understanding the shapes and patterns represented by equations in both polar and Cartesian systems. This process is crucial for interpreting mathematical data and applying it in real-world scenarios.
In our example, we started with the polar equation \( r \cos \theta = 2 \). By converting this to the Cartesian coordinate system, we obtained \( x = 2 \), which we identified as a graph of a vertical straight line.

• Vertical line graphs in the Cartesian coordinate system are characterized by the equation \( x = c \), where \( c \) is a constant.
• The line \( x = 2 \) means that for any value of \( y \), the value of \( x \) remains 2, forming a line parallel to the y-axis.

Graph identification allows us to swiftly interpret the nature and behavior of various geometric shapes derived from equations, offering insights into potential applications and implications of those graphs.