Problem 57 - Algebra and Trigonometry

Question

Convert the polar equation to rectangular coordinates. $$ r=2 \sec \theta $$

Step-by-Step Solution

Answer

The rectangular equation is \( x = 2 \).

Step 1: Understand the Polar Equation

We start with the polar equation \( r = 2 \sec \theta \). Recall that \( \sec \theta = \frac{1}{\cos \theta} \). Hence, the equation can be rewritten as \( r = \frac{2}{\cos \theta} \).

Step 2: Convert Secant to Cartesian Coordinates

Multiply both sides of the equation by \( \cos \theta \) to eliminate the fraction: \( r \cos \theta = 2 \). In polar coordinates, \( r \cos \theta = x \). So the equation becomes \( x = 2 \).

Step 3: Identify the Rectangular Equation

The equation \( x = 2 \) is already in rectangular form. It represents a vertical line on the Cartesian plane where every point on the line has an x-coordinate equal to 2.

Key Concepts

Polar Coordinates

Polar coordinates are a way to describe a point in a two-dimensional space using a distance and an angle. Instead of specifying a point by how far along and how far up it is, you'll denote it by:

• the distance from the origin (often noted as \( r \))
• and the angle made with the positive x-axis (often noted as \( \theta \))

For example, a point can be specified as \( (r, \theta) \). This system is especially useful in situations where you are dealing with angles and distances simultaneously, like in circular or spiral patterns. It can make calculations simpler than using traditional rectangular (Cartesian) coordinates. In the context of the given problem, we are dealing with the equation \( r = 2 \sec \theta \), which is presented in polar form. By understanding the basics of polar coordinates, we learn that the distance depends on the secant of the angle in this equation.

Rectangular Coordinates

Rectangular coordinates are probably the most familiar way to locate a point on a two-dimensional plane. This system is based on the concept of using two perpendicular lines known as the x-axis and y-axis. A point is expressed as \( (x, y) \), where:

• \( x \) is the horizontal distance from the origin (0,0)
• \( y \) is the vertical distance from the origin

In the solution provided, the polar equation was successfully converted into rectangular form. This involved recognizing that in polar coordinates, \( r \cos \theta = x \) and \( r \sin \theta = y \). By applying these relationships, the example equation simplifies to \( x = 2 \), which is a simple vertical line in the rectangular coordinate system.The result demonstrates how a complex-looking polar equation can translate to a straightforward equation in rectangular coordinates, making visualization easier on the Cartesian plane.

Cartesian Plane

The Cartesian plane, or rectangular coordinate system, provides a simple visual framework that helps us plot complex mathematical functions. It consists of two number lines: the x-axis running horizontally, and the y-axis running vertically. These axes intersect at the origin point (0,0), dividing the plane into four quadrants.In this plane, each point is determined uniquely by its horizontal (x) and vertical (y) coordinates. This system allows us to plot equations, draw shapes, and analyze spatial relationships with ease. Understanding this plane is crucial for interpreting equations in rectangular coordinates.In the exercise example, the equation \( x = 2 \) provides us with a clear line on the Cartesian plane. This line is vertical, crossing the x-axis exactly at the point where x is 2, regardless of the y-value. The Cartesian plane allows us to see this easily and interpret how changes in the polar equation affect its rectangular counterpart, showing us the power of coordinate transformations.