Understanding Polar Coordinates

While the rectangular (Cartesian) coordinate system uses a grid of horizontal and vertical lines to locate points as $(x, y)$, the polar coordinate system uses distances and angles. This system is especially useful for graphing curves like circles, spirals, and rosettes that are complicated to express in $x$ and $y$.

What are Polar Coordinates?

In the polar coordinate system, every point on a plane is defined by an ordered pair $(r, \theta)$:

• $r$ (radius): The directed distance from the origin (often called the pole).
• $\theta$ (angle): The directed angle measured counterclockwise from the positive $x$-axis (the polar axis).

Converting Between Systems

Because polar and rectangular coordinates can describe the exact same point on a plane, we can easily convert between them using right triangle trigonometry.

Imagine a point $P$ with rectangular coordinates $(x, y)$ and polar coordinates $(r, \theta)$. Drawing a line from the origin to $P$ creates a right triangle where the hypotenuse is $r$, the horizontal leg is $x$, and the vertical leg is $y$.

Polar to Rectangular To find $(x, y)$ when you know $(r, \theta)$: $x = r \cos \theta$ $y = r \sin \theta$

Rectangular to Polar To find $(r, \theta)$ when you know $(x, y)$: $r = \sqrt{x^2 + y^2}$ $\tan \theta = \frac{y}{x}$ (Note: Always check which quadrant your point $(x, y)$ is in to ensure you find the correct angle $\theta$.)

Example 1: Rectangular to Polar

Problem: Convert the rectangular coordinates $(3, 4)$ to polar coordinates.

1. Find $r$: $r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
2. Find $\theta$: Since $(3, 4)$ is in the first quadrant, we just evaluate the inverse tangent: $\tan \theta = \frac{4}{3}$ $\theta = \arctan\left(\frac{4}{3}\right) \approx 0.93 \text{ radians} \text{ (or } 53.1^\circ\text{)}$

Answer: The polar coordinates are approximately $(5, 0.93)$.

Example 2: Polar to Rectangular

Problem: Convert the polar coordinates $\left(5, \frac{\pi}{3}\right)$ to rectangular coordinates.

1. Find $x$: $x = 5 \cos\left(\frac{\pi}{3}\right) = 5 \left(\frac{1}{2}\right) = \frac{5}{2}$
2. Find $y$: $y = 5 \sin\left(\frac{\pi}{3}\right) = 5 \left(\frac{\sqrt{3}}{2}\right) = \frac{5\sqrt{3}}{2}$

Answer: The rectangular coordinates are $\left(\frac{5}{2}, \frac{5\sqrt{3}}{2}\right)$.