Problem 90 - College Trigonometry

Question

Convert the equation from polar coordinates into rectangular coordinates. $$ r=-2 \sec (\theta) $$

Step-by-Step Solution

Answer

Use the rectangular form of \(\sec(\theta)\) and simplify to get \(x = -2\).

Step 1: Convert using sec(θ) = 1/cos(θ)

\(r = -2\sec\theta = \frac{-2}{\cos\theta}\), so \(r\cos\theta = -2\).

Step 2: Convert to rectangular

Since \(r\cos\theta = x\): \(x = -2\). This is a vertical line.

Key Concepts

Polar Coordinates

Polar coordinates are a system for identifying the location of a point on a plane using two values: radius and angle. The radius, usually labeled as \( r \), represents the distance from the point to a fixed origin. The angle, designated as \( \theta \), measures the direction of the point relative to a fixed line, often the positive x-axis. This system is particularly useful for dealing with scenarios where relationships are more naturally expressed in terms of angles and distances from a central point.

In everyday situations, polar coordinates are often used when dealing with circular and rotational systems. For instance, describing the movement of planets around the sun, or tracking the path of a rotating fan blade, would naturally benefit from polar coordinates. This method of coordinate representation is easy to understand when you're dealing with curves or rotations, yet it can be converted into a form more suitable for calculations, like rectangular coordinates.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use two perpendicular axes to pinpoint the position of a point. These axes are typically labeled as the x-axis and y-axis. In this system, every point on a plane is defined by a pair of numbers \((x, y)\). These numbers represent the horizontal and vertical distances from the origin to the point. Rectangular coordinates are incredibly useful for everyday tasks, such as graphing equations, solving geometry problems, or mapping locations on a grid.

One of the primary advantages of rectangular coordinates is their simplicity and directness. By working directly with horizontal and vertical distances, it becomes straightforward to perform algebraic operations and geometric transformations. Despite their usefulness, rectangular coordinates can sometimes become cumbersome with circular or radial patterns, which is where polar coordinates come in handy.

Coordinate Transformations

Coordinate transformations are crucial for converting one form of coordinates into another. This process is essential in mathematical applications, allowing us to take advantage of the strengths of both polar and rectangular coordinate systems. Transformations are achieved by using a set of relationships between the two systems.

When converting from polar to rectangular coordinates, we rely on equations that express \( x \) and \( y \) in terms of the polar coordinates \( r \) and \( \theta \):

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

These equations allow us to determine the exact position within a plane in rectangular form. Conversely, to convert from rectangular to polar coordinates, you use:

• \( r = \sqrt{x^2 + y^2} \)
• \( \theta = \arctan(\frac{y}{x}) \)

Understanding these equations makes converting between the two forms not just possible, but straightforward, offering more flexibility and accuracy in problem-solving.