Lesson 10.3 : Polar Coordinates

New Concept

Polar coordinates describe a point's location using a distance from the origin, $r$, and an angle from an axis, $\theta$. You'll learn to plot points, convert coordinates like $(r, \theta)$ to $(x, y)$, and transform entire equations between these systems.

What’s next

Next, you'll dive into interactive examples for plotting points and converting coordinates. Then, test your skills with a series of practice problems.

Property

In the polar coordinate system, points are labeled $(r, \theta)$ and plotted on a polar grid. The grid consists of concentric circles radiating from the pole (the origin).
The first coordinate, $r$, is the radius or length of the directed line segment from the pole. The second coordinate, $\theta$, is the angle in radians measured counterclockwise from the polar axis (the positive x-axis).
To plot a point, rotate by the angle $\theta$ and then move a distance $r$ from the pole. If $r$ is negative, move in the direction opposite to the angle.

Examples

• To plot the point $(4, \frac{\pi}{3})$, you rotate counterclockwise by $\frac{\pi}{3}$ radians (or 60 degrees) from the polar axis and move 4 units out from the pole.
• To plot the point $(3, -\frac{\pi}{4})$, you rotate clockwise by $\frac{\pi}{4}$ radians (or 45 degrees) and move 3 units out from the pole.
• To plot the point $(-2, \frac{\pi}{6})$, you first face the $\frac{\pi}{6}$ direction, but because $r$ is negative, you move 2 units in the exact opposite direction, into the third quadrant.

Explanation

Think of polar coordinates as giving directions. The angle $\theta$ tells you which way to face from the starting point (the pole), and the radius $r$ tells you how many steps to take. A negative $r$ just means you walk backward!

Property

To convert polar coordinates $(r, \theta)$ to rectangular coordinates $(x, y)$, use the following formulas derived from the relationships in a right triangle:

Given a polar coordinate $(r, \theta)$, first evaluate $\cos \theta$ and $\sin \theta$. Then, multiply each result by $r$ to find the corresponding $x$ and $y$ coordinates.

Property

To convert rectangular coordinates $(x, y)$ to polar coordinates $(r, \theta)$, use the following relationships:

When finding $\theta$, you must consider the quadrant of the point $(x, y)$ to ensure the angle is correct. A single rectangular point can have multiple equivalent polar representations.

Property

To write a Cartesian equation in polar form, eliminate the variables $x$ and $y$ by using the relationships $x = r \cos \theta$ and $y = r \sin \theta$. A common and useful substitution is replacing $x^2 + y^2$ with $r^2$. The goal is typically to write the equation with $r$ as a function of $\theta$.

Examples

• Convert $x^2 + y^2 = 16$ to polar form. Substitute $x^2 + y^2$ with $r^2$ to get $r^2 = 16$, which simplifies to $r = 4$.
• Convert $x^2 + y^2 = 10y$ to polar form. Substitute $r^2$ for $x^2 + y^2$ and $r \sin \theta$ for $y$. This gives $r^2 = 10r \sin \theta$. Dividing by $r$ (assuming $r ≠ 0$) gives $r = 10 \sin \theta$.
• Convert the line $y = 5x$ to polar form. Substitute $y = r \sin \theta$ and $x = r \cos \theta$ to get $r \sin \theta = 5r \cos \theta$. This simplifies to $\tan \theta = 5$, or $\theta = \arctan(5)$.

Explanation

We can translate entire equations, not just points. This involves swapping the rectangular variables ($x$ and $y$) for their polar equivalents. The most powerful move is turning the expression for a circle's equation, $x^2 + y^2$, into a simple $r^2$.

Property

To rewrite a polar equation in Cartesian form, eliminate $r$ and $\theta$ by using the substitutions $r^2 = x^2 + y^2$, $x = r \cos \theta$, and $y = r \sin \theta$. It is often necessary to manipulate the equation first, such as by multiplying by $r$ or squaring both sides, to create expressions that can be easily substituted.

Examples

• Convert $r = 5 \sin \theta$ to Cartesian form. Multiply both sides by $r$ to get $r^2 = 5r \sin \theta$. Substitute $r^2 = x^2 + y^2$ and $y = r \sin \theta$ to get $x^2 + y^2 = 5y$.
• Convert $r = 4 \sec \theta$ to Cartesian form. First, rewrite $\sec \theta$ as $\frac{1}{\cos \theta}$, so $r = \frac{4}{\cos \theta}$. This gives $r \cos \theta = 4$. Since $x = r \cos \theta$, the equation is $x = 4$.
• Convert the equation $r = \frac{2}{1 - \cos \theta}$ to Cartesian form. Multiply by $1 - \cos \theta$ to get $r - r \cos \theta = 2$. Substitute $r = \sqrt{x^2+y^2}$ and $x = r \cos \theta$ to get $\sqrt{x^2+y^2} - x = 2$.

Explanation

This process converts polar graphs back into familiar rectangular forms like lines, circles, or other shapes. The key is to strategically manipulate the equation to create terms like $r^2$ or $r \cos \theta$ that you can replace with their $x$ and $y$ versions.