# 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:

1. 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.
2. 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:

1. From Cartesian to Polar:

• $r=\sqrt{{x}^{2}+{y}^{2}}$
• $\theta =\mathrm{arctan}\left(\frac{y}{x}\right)$
1. From Polar to Cartesian:

• $x=r\cdot \mathrm{cos}\left(\theta \right)$
• $y=r\cdot \mathrm{sin}\left(\theta \right)$

IV. Common Polar Graphs:

1. Circle:

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

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

• $r=a\cdot \mathrm{cos}\left(n\theta \right)$ or $r=a\cdot \mathrm{sin}\left(n\theta \right)$, where $a$ is a constant and $n$ is an integer.

V. Polar Equations for Conic Sections:

1. Circle:

• $r=a$
2. Ellipse:

• $r=\frac{a\left(1-{e}^{2}\right)}{1-e\cdot \mathrm{cos}\left(\theta \right)}$, where $e$ is the eccentricity.
3. Hyperbola:

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

VI. Area and Length in Polar Coordinates:

1. Area:

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

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