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 $x>0,$ then the point $(x, y)$ on the rectangular coordinate system can be represented by $(r, \theta)$ on the polar coordinate system, where $r=\sqrt{x^{2}+y^{2}}$ and $\theta=\arctan (y / x)$

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 $x>0,$ then the point $(x, y)$ on the rectangular coordinate system can be represented by $(r, \theta)$ on the polar coordinate system, where $r=\sqrt{x^{2}+y^{2}}$ and $\theta=\arctan (y / x)$

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Key Concepts

Principal Value of the Arctan Function

The function arctan(y/x) returns the principal value of the angle in the interval (–?/2, ?/2). When x is positive, this value directly corresponds to the angle in polar coordinates. Understanding the domain and range of inverse trigonometric functions is crucial to applying these conversion formulas correctly.

Conversion Between Coordinate Systems

The conversion between rectangular and polar coordinates is achieved using the formulas r = ?(x² + y²) for the radial distance and ? = arctan(y/x) for the angle. These relationships allow one to switch representations of a point in the plane, facilitating the solution of problems that are more naturally expressed in one system over the other.

Rectangular Coordinates

Rectangular coordinates describe a point in the plane by its horizontal and vertical distances (x, y) from a fixed origin. This coordinate system is foundational to Euclidean geometry and serves as the basis for many algebraic and geometric methods.

Polar Coordinates

Polar coordinates represent a point in the plane by a distance from the origin (r) and an angle (?) measured from the positive x-axis. This system is particularly useful for problems with circular symmetry and provides an alternative view to the rectangular system.

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