Problem 90 - Algebra and Trigonometry

Question

If you are given polar coordinates of a point, explain how to find two additional sets of polar coordinates for the point.

Step-by-Step Solution

Answer

Given a point with polar coordinates (r, θ), two additional sets of polar coordinates for the same point can be found as (r, θ + 2nπ) and (-r, θ + π + 2nπ), where n is any integer.

Step 1: Overview

Given a point with polar coordinates (r, θ), we are to find two additional pairs of polar coordinates that represent the same point.

Step 2: Adding 2π to the angle

The first additional set of polar coordinates can be obtained by adding or subtracting an integer multiple of \( 2\pi \) from the angle \( \theta \). Let's denote \( n \) as any whole number. The first new set of polar coordinates can therefore be defined as \( (r, θ + 2n\pi) \). Because a complete rotation about the polar axis is \( 2\pi \) radians, adding or subtracting \( 2n\pi \) does not change the location of the point.

Step 3: Alternating the radius and the angle

Another possible pair can be found by alternating the sign of \( r \) and subsequently changing \( \theta \) by \( \pi \) radians. That's because in polar coordinates, positive \( r \) signifies the distance of the point from origin in direction specified by \( \theta \) but negative \( r \) means direction is opposite to \( \theta \). So, the second additional set of polar coordinates could be defined as \( (-r, θ + \pi + 2n\pi) \).