Problem 78 - College Trigonometry

Question

Convert the equation from polar coordinates into rectangular coordinates. $$ r=-3 $$

Step-by-Step Solution

Answer

The rectangular form is \(x^2 + y^2 = 9\).

Step 1: Understand Polar to Rectangular Conversion

In polar coordinates, points are described using the distance from the origin \(r\) and an angle \(\theta\). Rectangular coordinates use \(x\) and \(y\). The conversion formulas are: \(x = r\cos\theta\) and \(y = r\sin\theta\).

Step 2: Identify Given Values

The given polar equation is \(r = -3\). Here, \(r\) is the distance from the origin. Since no specific angle \(\theta\) is provided in the problem statement, we consider all possibilities.

Step 3: Convert Using Polar to Rectangular Formulas

Without a specific \(\theta\), focus on the meaning of \(r = -3\). This represents a circle with radius 3 centered at the origin but negated (pointing radially inward). Thus the equation in rectangular coordinates is \(x^2 + y^2 = (-3)^2\).

Step 4: Simplify Rectangular Equation

Simplifying the circle equation gives \(x^2 + y^2 = 9\). This represents a circle of radius 3 centered at the origin in the rectangular coordinate system.

Key Concepts

Polar Coordinates

In polar coordinates, the location of a point is determined by two values: the radius, denoted as \( r \), and the angle, referred to as \( \theta \). These coordinates are particularly useful for problems that involve circles and rotations, as they naturally express distance from a central point and direction.
Polar coordinates are generally represented as \( (r, \theta) \), where:

• \( r \) is the distance from the origin (or pole).
• \( \theta \) is the angle measured counterclockwise from the positive x-axis.

The polar system offers a powerful way to describe circular motion and paths. Moving around the circle simply changes \( \theta \) while keeping \( r \) constant.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use an x-y plane to locate points. Each point is determined by how far it is along the x-axis and the y-axis from an origin.
These coordinates are expressed as \( (x, y) \):

• \( x \) represents the horizontal position.
• \( y \) represents the vertical position.

Rectangular coordinates are ideal for measuring straight-line distances and solving algebraic equations. They form the basis of most introductory physics and engineering calculations.

Conversion Formulas

Conversion between polar and rectangular coordinates bridges between different methods of describing positions. The conversion formulas are crucial to translate problems into the system that best suits their nature.
To convert from polar to rectangular coordinates, use the following relationships:

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

These formulas derive from trigonometry, considering the projection of the radius onto the x and y axes. When \( r \) is negative, it points in the opposite direction from the angle, affecting the resulting \( x \) and \( y \) values. Understanding these conversions is key to interpreting and solving geometric figures in different coordinate systems.

Circle Equation

The equation of a circle in rectangular coordinates provides a straightforward representation of a circle's shape and size. For a circle centered at the origin, the equation is:

1. \( x^2 + y^2 = r^2 \)

Where \( r \) is the radius of the circle. The given problem starts with a polar equation \( r = -3 \).
Recognizing that \( r \) represents the radius in a direction, using the conversion formula led to the simple circle equation \( x^2 + y^2 = 9 \), representing a standard circle with radius 3 centered at the origin in the x-y plane.
This shows that regardless of direction, the radius square value determines the relative size of the circle. The circular symmetry remains unchanged whether the radius is signed positively or negatively.