Polar Coordinates
I. Introduction: Polar coordinates provide an alternative system for locating points in a plane. Unlike Cartesian coordinates (x, y), polar coordinates use a radial distance (r) and an angle (θ) to represent a point. Understanding polar coordinates is crucial in various fields, including physics, engineering, and mathematics.
II. Representation of Polar Coordinates:

Basic Definitions:
 Polar Axis: The reference line (usually horizontal) from which the angle is measured.
 Pole (Origin): The point where the polar axis intersects the plane.

Coordinate Notation:
 A point in polar coordinates is represented as (r, θ), where r is the radial distance from the pole, and θ is the angle measured in a counterclockwise direction from the polar axis.
III. Conversion between Cartesian and Polar Coordinates:

From Cartesian to Polar:
 $r=\sqrt{{x}^{2}+{y}^{2}}$

 $\theta =\mathrm{arctan}\left(\frac{y}{x}\right)$

From Polar to Cartesian:
 $x=r\cdot \mathrm{cos}(\theta )$
 $y=r\cdot \mathrm{sin}(\theta )$
IV. Common Polar Graphs:

Circle:
 The equation of a circle in polar coordinates is $r=a$, where $a$ is the radius.

Line:
 The equation of a line through the pole is $\theta =\alpha $, where $\alpha $ is the angle with the polar axis.

Rose Curves:
 $r=a\cdot \mathrm{cos}(n\theta )$ or $r=a\cdot \mathrm{sin}(n\theta )$, where $a$ is a constant and $n$ is an integer.
V. Polar Equations for Conic Sections:

Circle:
 $r=a$

Ellipse:
 $r=\frac{a(1{e}^{2})}{1e\cdot \mathrm{cos}(\theta )}$, where $e$ is the eccentricity.

Hyperbola:
 $r=\frac{a(1+{e}^{2})}{1+e\cdot \mathrm{cos}(\theta )}$, where $e$ is the eccentricity.
VI. Area and Length in Polar Coordinates:

Area:
 The area $A$ enclosed by a polar curve $r=f(\theta )$ between two angles $\alpha $ and $\beta $ is given by $A=\frac{1}{2}{\int}_{\alpha}^{\beta}[f(\theta ){]}^{2}\text{\hspace{0.17em}}d\theta $.

Arc Length:
 The length $L$of a polar curve $r=f(\theta )$ between two angles $\alpha $ and $\beta $ is given by $L={\int}_{\alpha}^{\beta}\sqrt{[r(\theta ){]}^{2}+{\left(\frac{dr}{d\theta}\right)}^{2}}\text{\hspace{0.17em}}d\theta $.