Calculus
Textbooks
Boundless Calculus
Differential Equations, Parametric Equations, and Sequences and Series
Parametric Equations and Polar Coordinates
Calculus Textbooks Boundless Calculus Differential Equations, Parametric Equations, and Sequences and Series Parametric Equations and Polar Coordinates
Calculus Textbooks Boundless Calculus Differential Equations, Parametric Equations, and Sequences and Series
Calculus Textbooks Boundless Calculus
Calculus Textbooks
Calculus
Concept Version 7
Created by Boundless

Polar Coordinates

Polar coordinates define the location of an object in a plane by using a distance and an angle from a reference point and axis.

Learning Objective

  • Use a polar coordinate to define a point with $r$ (distance from pole), and $\theta$(angle between axis and ray)


Key Points

    • Polar coordinates use a distance from a central point called a radial distance, usually specified as $r$.
    • Polar coordinates use an angle measurement from a polar axis, which is usually positioned as horizontal and pointing to the right. Counterclockwise is usually positive.
    • To convert from Polar coordinates to Cartesian coordinates, draw a triangle from the horizontal axis to the point. The $r$ coordinate is $r \cos \theta$ and the y coordinate is $r \sin \theta$.

Term

  • polar

    of a coordinate system, specifying the location of a point in a plane by using a radius and an angle


Full Text

We use coordinate systems every day, even if we don't realize it. For example, if you walk 20 meters to the right of the parking lot to find the car, you are using a coordinate system. Coordinate systems are a way of determining the location of a point or object of interest in relation to something else. The coordinate system you are most likely familiar with is the $xy$-coordinate system, where locations are described as horizontal ($x$) and vertical ($y$) distances from an arbitrary point. This is called the Cartesian coordinate system.

The $xy$ or Cartesian coordinate system is not always the easiest system to use for every problem. In certain problems, like those involving circles, it is easier to define the location of a point in terms of a distance and an angle. Such definitions are called polar coordinates.

In polar coordinates, each point on a plane is defined by a distance from a fixed point and an angle from a fixed direction.

The distance is known as the radial distance and is usually denoted as $r$.

The angle is known as the polar angle, or radial angle, and is usually given as $\theta$.

A positive angle is usually measured counterclockwise from the polar axis, and a positive radius is in the same direction as the angle. A negative radius would be opposite the direction of the angle and a negative angle would be measured clockwise from the polar axis. The polar axis is usually drawn horizontal and pointing to the right .

Polar Coordinates

A set of polar coordinates. Note the polar angle increases as you go counterclockwise around the circle with 0 degrees pointing horizontally to the right.

Relation between Cartesian and Polar Coordinates

The $x$ Cartesian coordinate is given by $r \cos \theta$ and the $y$ Cartesian coordinate is given by $r \sin \theta$.

Polar coordinates in $r$ and $\theta$ can be converted to Cartesian coordinates $x$ and $y$. This can be done by noting that the radial distance $r$ and the polar angle $\theta$ can define a triangle with a horizontal length $x$ and vertical length $y$ . Thus, using trigonometry, it can be shown that the $x$ coordinate is $r \cos \theta$ and the $y$ coordinate is $r \sin \theta$.

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