\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus - Polar Coordinates


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Polar coordinates is one of those topics that can be taught in many different courses. Some students come across the topic in physics for the first time. Sometimes, it's in precalculus or trig. No matter what course you are in right now, you will find everything you need here on polar coordinates. If you've already learned this topic, take some time to review the material on these pages and watch a few videos anyway, so that it is fresh in your mind.

Basic Idea of Polar Coordinates

You should already be familiar with graphing in rectangular coordinates (sometimes called cartesian coordinates). We can use trigonometry to describe the same point(s) or graph another way, as shown in plot 1.

From basic trig, you know that a point in the plane can be described as \((x,y)\) or as \(( r \cos(\theta), r \sin(\theta) ) \). Comparing these two forms gives you the equations

\( x = r \cos(\theta) \)


\( y = r \sin(\theta) \)

These equations are used to convert between polar coordinates and rectangular coordinates.

Remember from trig that angles can be described in an infinite number of ways, since \( \theta = \theta + 2\pi \) and \(\theta = \theta - 2\pi\). It is always best to use the smallest possible angle in the interval \( (-\pi, \pi]\) or \( [0, 2\pi) \) or whatever is required by the context.

One of the biggest differences you will find between trig and polar coordinates is that in trig, r in the above equation is usually \(1\). Trig focuses on the unit circle (when \(r=1\)). However, in polar coordinates we generalize the equations so that r is usually not \(1\).

The positive x-axis is called the polar axis, labeled L in plot 2 and the point O is called the pole. All angles are measured from the polar axis with positive angles in a counter-clockwise direction.

Polar coordinates are just parametric equations where the parameter is the angle \(\theta\) and r is a function of \(\theta\). It will help you to understand polar coordinates if you have a good understanding of parametrics. Go to the parametrics section for more information.

Note: In plot 2, the angles are measured in degrees. However, in calculus we almost always specify angles in radians. To convert between radians and degrees, remember that \( 2\pi = 360^o \).

Okay, it's time to watch some videos. This first video is really good to give you an overview of polar coordinates.

Krista King Math - Overview of Polar Coordinates [7min-24secs]

video by Krista King Math

Here is another good video introduction to polar coordinates. He uses graphs and examples very effectively in this video.

PatrickJMT - Introduction to Polar Coordinates [10min-34secs]

video by PatrickJMT

Okay, now you know the basics of polar coordinates and you can work most problems you come across. However, if you really want to understand polar coordinates, then this video clip is good to watch. It gives a more in-depth discussion with some very good examples, some unique, which will help you a lot.

MIT OCW - Understand Polar Coordinates [20min-35secs]

The first half of this video discusses parametric equations but you don't need that material to understand the second half on polar coordinates.

video by MIT OCW

Graphing In Polar Coordinates

Krista King Math - How do you sketch a polar curve (cardioid)? [6min-32secs]

video by Krista King Math


With the basics of polar coordinates under your belt, now it is time to work directly with the points and equations to convert between rectangular and polar coordinates. You will find discussion, videos and practice problems on the next page.

polar coordinates 17calculus youtube playlist

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Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\)

\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\)

\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\)

\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\)

Set 2 - squared identities

\( \sin^2t + \cos^2t = 1\)

\( 1 + \tan^2t = \sec^2t\)

\( 1 + \cot^2t = \csc^2t\)

Set 3 - double-angle formulas

\( \sin(2t) = 2\sin(t)\cos(t)\)

\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\)

Set 4 - half-angle formulas

\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\)

\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\)

Trig Derivatives

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\)


\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\)

\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\)


\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\)

\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\)


\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\)

Inverse Trig Derivatives

\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\)


\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\)

\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\)


\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\)

\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\)


\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

Trig Integrals

\(\int{\sin(x)~dx} = -\cos(x)+C\)


\(\int{\cos(x)~dx} = \sin(x)+C\)

\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)


\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)

\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\)


\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\)

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Basic Idea of Polar Coordinates

Graphing In Polar Coordinates


next: converting to/from polar coordinates →

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