Polar coordinates

Definition

Polar coordinates represent a point in a plane using two values: the radial distance from a reference point (origin) and the angle from a reference direction. They are useful for problems involving circular or rotational symmetry.

5 Must Know Facts For Your Next Test

1. Polar coordinates are denoted as $(r, \theta)$, where $r$ is the radius and $\theta$ is the angle.
2. $r$ must be non-negative, while $\theta$ can take any real value, typically measured in radians.
3. Conversion between polar and Cartesian coordinates involves $x = r \cos(\theta)$ and $y = r \sin(\theta)$. Conversely, $r = \sqrt{x^2 + y^2}$ and $\theta = \tan^{-1}(y/x)$.
4. Polar coordinates simplify the representation of vectors that originate from or pass through the origin in circular motion problems.
5. In physics, polar coordinates are often used to solve problems involving oscillations, waves, and rotational dynamics.

Review Questions

• How do you convert Cartesian coordinates $(x, y)$ to polar coordinates $(r, \theta)$?
• What are the advantages of using polar coordinates in problems involving rotational symmetry?
• Explain how you would represent a vector using polar coordinates.

Related terms

Cartesian Coordinates: A coordinate system that specifies each point uniquely by a pair of numerical coordinates $(x,y)$.

Vector Components: Parts of a vector that show its influence along specific axes; in Cartesian systems these are usually represented as $x$ and $y$ components.

Radians: A unit of angular measure used in mathematics; one radian is the angle subtended by an arc whose length is equal to its radius.