Problem 104 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When converting a point from polar coordinates to rectangular coordinates, there are infinitely many possible rectangular coordinate pairs.

Step-by-Step Solution

Answer

The statement does not make sense, because while there can be infinitely many polar coordinates for the same point due to the periodicity of theta, there is only one unique rectangular coordinate for each unique point.

Step 1: Understanding Polar and Rectangular Coordinate Systems

In Polar coordinates, a location in a plane is determined by a distance from a reference point (r, the radial coordinate) and an angle from a reference direction (theta, the angular coordinate). This contrasts with Rectangular coordinates which identify the location of a point by its distance along the x and y axes, away from the origin (x,y).

Step 2: Transformation from Polar to Rectangular Coordinates

When transforming polar coordinates to rectangular coordinates, we use the formulas \(x = r \cos(\theta)\) and \(y = r \sin(\theta)\). These formulas yield only a single pair of rectangular coordinates for any given polar coordinates.

Step 3: Reasoning the Statement

Hence the statement 'When converting a point from polar coordinates to rectangular coordinates, there are infinitely many possible rectangular coordinate pairs' does not make sense. Each unique point in the polar coordinate system will correspond to one unique point in the rectangular coordinate system, despite the fact that there are infinitely many polar coordinates for a given point due to the periodic nature of the angle theta.