Question

Convert the polar equation to a rectangular equation. \(r=\frac{3}{\cos \theta}\)

Short answer

The rectangular equation is \( x = 3 \).

Step-by-step solution

Understand Polar and Rectangular Coordinates

Polar coordinates describe a point by the distance from the origin \(r\) and the angle \(\theta\) from the positive x-axis. Rectangular coordinates (Cartesian coordinates) describe a point using \(x\) and \(y\) coordinates in the plane.

Identify the Polar Equation

The given polar equation is \( r = \frac{3}{\cos \theta} \).

Key concepts

Polar Coordinates

In mathematics, polar coordinates are a way of describing points in a plane using two pieces of information. First, they specify the distance from a fixed point, often called the origin. This distance is represented by the variable \( r \). Second, they utilize an angle \( \theta \) that is measured from a fixed direction, usually from the positive x-axis.

Consider polar coordinates as an alternative to the more familiar rectangular (or Cartesian) coordinates. Unlike in the rectangular system where a point is located by moving right or left and up or down, in polar coordinates, you think about moving in a circular path around the origin. This system is particularly helpful in contexts where angles and distances play a significant role, like in navigation or circular motion.

• The radius \( r \) indicates how far away you are from the center of the system.
• The angle \( \theta \) shows the direction from the origin to your point.

Understanding this perspective aids in converting between coordinate systems as seen in the exercise, where the relationship between \( r \) and \( \theta \) is crucial for conversion.

Rectangular Coordinates

Unlike polar coordinates, rectangular coordinates frame points in a plane using two perpendicular lines intersecting at the origin, forming what is known as the x and y axes. Each point is specified by how far it is to the right or left of the y-axis and how far it is above or below the x-axis.

In rectangular coordinates, a point \((x, y)\) is plotted by moving \(x\) units horizontally and \(y\) units vertically. This makes the system extremely easy to use for plotting graphs and performing algebraic manipulations, which explains its popularity in geometry and algebra.

• The x-coordinate tells you the position along the horizontal direction.
• The y-coordinate specifies the vertical position.

This makes it straightforward for drawing straight lines and mapping shapes onto a grid. In the exercise, converting from polar to rectangular coordinates involves relating these \(x\) and \(y\) values in terms of \( r \) and \( \theta \). Specifically, you can use:

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

Coordinate Systems

A coordinate system provides a consistent way to identify the location of points in space. Both the polar and rectangular systems serve as distinct, yet interrelated, methods of expressing spatial positions.

Understanding their differences and the process of conversion between them is essential because each system has its own strengths. In some situations, using polar coordinates might simplify the equations you are dealing with, especially if you are working with curves that pivot around a central point. Rectangular coordinates, however, are more intuitive for describing shapes and lines that interact linearly.

• Polar and rectangular systems are interconnected through the relationships: \( x = r \cos \theta \), \( y = r \sin \theta \), and \( r = \sqrt{x^2 + y^2} \).
• Conversion between coordinate systems often involves understanding trigonometry, as it helps transition between circular and linear perspectives.

Recognizing how these coordinate systems translate between the representation of the same point provides the flexibility needed to approach problems optimally, particularly in exercises like the given one, where translating between forms facilitates new insights into the equations at hand.