True or False? In Exercises 113-116, determine whether the...

True or False? In Exercises $113-116,$ determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If $\left(r_{1}, \theta_{1}\right)$ and $\left(r_{2}, \theta_{2}\right)$ represent the same point on the polar coordinate system, then $\left|r_{1}\right|=\left|r_{2}\right|$

True or False? In Exercises $113-116,$ determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.
If $\left(r_{1}, \theta_{1}\right)$ and $\left(r_{2}, \theta_{2}\right)$ represent the same point on the polar coordinate system, then $\left|r_{1}\right|=\left|r_{2}\right|$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Polar Coordinate System

The polar coordinate system represents points in the plane by using a distance from a fixed origin (called the radial coordinate) and an angle from a fixed direction (the angular coordinate). This system often gives rise to multiple representations for the same point due to its inherent periodicity in angle and the allowance of negative radii.

Equivalence of Polar Coordinates

Two polar coordinates represent the same point if one of two conditions is satisfied: either their radii are the same and their angles differ by an integer multiple of 2?, or one radius is the negative of the other and the corresponding angles differ by an odd multiple of ?. This equivalence relation is central to understanding the non-unique representation of points in polar coordinates.

Absolute Value

The absolute value of a number is a measure of its magnitude, disregarding its sign. In the context of polar coordinates, the absolute value of the radial coordinate represents the actual distance from the origin. This concept ensures that even when the radial coordinate is negative, its absolute value correctly reflects the distance of the point from the origin.

Transcript

Please give Ace some feedback