True or False If r1 and r2 are not 0, and if (r1, θ) and (r2, θ+π) represent the same point in t

step 1 So our problem tells us that these two points are going to actually correspond to the same point

True or False If r1 and r2 are not 0, and if (r1, θ) and...

True or False If $r_{1}$ and $r_{2}$ are not $0,$ and if $\left(r_{1}, \theta\right)$ and $\left(r_{2}, \theta+\pi\right)$ represent the same point in the plane, then $r_{1}=-r_{2} .$ Justify your answer.

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Key Concepts

Polar Coordinates

This is a coordinate system in which points on a plane are defined by a distance from a fixed reference point (the origin) and an angle from a fixed direction. It contrasts with Cartesian coordinates by emphasizing a point's direction and distance rather than its horizontal and vertical displacements.

Equivalence of Polar Representations

In polar coordinates, a single point can be represented in multiple ways. A point expressed as (r, ?) is equivalent to (r, ? + 2?k) for any integer k. This equivalence arises because adding any full rotation (360° or 2? radians) to the angle leaves the point's position unchanged.

Negative Radius Representation

A negative value for the radius in polar coordinates indicates that the point lies in the direction opposite to the angle provided. Specifically, the point (?r, ?) is equivalent to the point (r, ? + ?), which demonstrates how a point can be represented with both a positive and a negative radius by appropriately adjusting the angle.

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