Defining Polar Coordinates (College Board AP® Calculus BC)

Study Guide

Mark Curtis

Written by: Mark Curtis

Reviewed by: Dan Finlay

Updated on

Defining polar coordinates

What are polar coordinates?

Diagram of polar coordinates with a point P(x, y) or (r, θ). The angle θ and radius r are marked from the origin, labelled as "Pole" with the x-axis as "Initial Line".
  • Polar coordinates are an alternative way to describe the position of points and curves in two-dimensions using

    • a distance, r

      • where r can be positive or negative

      • measured from an origin, called a pole

    • and an angle, theta

      • measured in radians from an initial line (extending from the pole)

      • The initial line is usually the x-axis

  • The polar coordinates of a point are written as open parentheses r comma space theta close parentheses

    • These are not to be confused with a point given in open parentheses x comma space y close parentheses form

  • Negative values of r are drawn in the opposite direction to theta

    • open parentheses negative r comma space theta close parentheses equals open parentheses r comma space theta plus pi close parentheses

Examiner Tips and Tricks

It will be clear in the question whether polar coordinates or x and y coordinates are being used.

  • If a point P has polar coordinates open parentheses r comma space theta close parentheses, then the x and y coordinates in the x y-plane are found by trigonometry:

    • x equals r space cos space theta

    • y equals r space sin space theta

  • It can also be helpful to make r squared and theta the subject to give two more relationships:

    • r squared equals x squared plus y squared

    • theta equals arctan open parentheses y over x close parentheses

  • These are found using trigonometric identities:

    • x squared plus y squared equals open parentheses r cos space theta close parentheses squared space plus space open parentheses r sin space theta close parentheses squared equals r squared open parentheses cos squared theta plus sin squared theta close parentheses equals r squared

    • and y over x equals fraction numerator r space sin space theta over denominator r space cos space theta end fraction equals tan space theta

How do I sketch a polar curve?

  • A polar curve is a curve given in the form r equals f open parentheses theta close parentheses

    • They are sometimes given in the form r squared equals g open parentheses theta close parentheses

  • The polar curve may be restricted to a range of theta values, e.g. 0 less or equal than theta less or equal than pi over 2

  • To sketch a polar curve, use the graph sketching function on your calculator

    • Change the type of graph to 'polar'

  • An alternative way to sketch is by plotting points

    • Find the distance r for some key angles, theta equals 0 comma pi over 6 comma pi over 3 comma space... space 2 pi

      • Plot these points on a polar grid (see below)

Polar coordinate graph with concentric circles and radial lines, displaying angles in radians, such as π/2, π, and 2π, with intersecting lines.
  • The following are some examples of polar curves (you do not need to learn these):

Diagram of eight polar graphs with labels: Circle, Cardioid, Limacons, Lemniscate, Rose Curves, and Archimedes' Spiral, showing equations.
  • Note that the spiral r equals k theta has an angle theta that can continue beyond 2 pi

How do I find x and y coordinates on a polar curve?

  • To find the x and y coordinates of a point on a polar curve, substitute the equation of the polar curve, r equals f open parentheses theta close parentheses, into the trigonometric relationships x equals r space cos space theta and y equals r space sin space theta to get:

    • x equals f open parentheses theta close parentheses space cos space theta

    • y equals f open parentheses theta close parentheses space sin space theta

  • Different values of theta will give different x and y coordinates

Examiner Tips and Tricks

The two relationships x equals r space cos space theta and y equals r space sin space theta are used a lot in harder polar questions.

How do I find the average distance from the origin to a point on a polar curve?

  • The average distance from the origin to a point on a polar curve, r open parentheses theta close parentheses, where alpha less or equal than theta less or equal than beta is:

fraction numerator 1 over denominator beta minus alpha end fraction integral subscript alpha superscript beta r open parentheses theta close parentheses space d theta

How do I convert a polar equation into (and out of) x and y coordinates?

  • To convert an equation from polar coordinates into x and y coordinates, use the algebraic relationships above

    • e.g. r equals 4 space cos space theta

      • Multiply both sides by r to get r squared equals 4 r space cos space theta

      • Use that r squared equals x squared plus y squared and that r space cos space theta equals x to give x squared plus y squared equals 4 x

      • By completing the square, this is open parentheses x minus 2 close parentheses squared plus y squared equals 2 squared

      • which is a circle, radius 2, centre open parentheses 2 comma space 0 close parentheses

    • Note that r squared equals x squared plus y squared means r equals plus-or-minus square root of x squared plus y squared end root

      • Only use the negative square root for parts of the curve where r goes negative

  • To convert an equation from x and y into polar coordinates, simply substitute in x equals r space cos space theta and y equals r space sin space theta and rearrange

    • You need to be able to recognize the following key examples:

x and y equation

Converting

Polar equation

y equals k

(horizontal line)

r space sin space theta equals k

r equals space k space csc space theta

x equals k

(vertical line)

r space cos space theta equals k

r equals k space sec space theta

x squared plus y squared equals a squared

(circle, centre O, radius a)

r squared cos squared space theta plus r squared sin squared space theta equals a squared

r equals a

Worked Example

The polar curve r equals sec squared space theta is shown below, where 0 less or equal than theta less or equal than pi over 4. The point P is on the curve at an angle of theta equals pi over 4.

Axes x and y axes shown, a curved polar line with a point P at the top labelled, and a dashed line forming a 45-degree angle (π/4) with the x-axis.

(a) Find the value of r at the point P.

Substitute theta equals pi over 4 into the equation r equals sec squared space theta to find r and simplify

r equals sec squared open parentheses pi over 4 close parentheses
equals 1 over open parentheses cos space pi over 4 close parentheses squared
equals 1 over open parentheses fraction numerator square root of 2 over denominator 2 end fraction close parentheses squared
equals 2

r equals 2

(b) Find the value of theta for which the x-coordinate of the curve is equal to fraction numerator 2 over denominator square root of 3 end fraction.

The x-coordinate of a polar curve is given by x equals r space cos space theta

First, substitute in x equals fraction numerator 2 over denominator square root of 3 end fraction

r space cos space theta equals fraction numerator 2 over denominator square root of 3 end fraction

Next, substitute in the polar curve r equals sec squared space theta for r, simplifying the left-hand side

table row cell sec squared space theta space cos space theta end cell equals cell fraction numerator 2 over denominator square root of 3 end fraction end cell row cell fraction numerator 1 over denominator cos squared space theta end fraction cross times cos space theta end cell equals cell fraction numerator 2 over denominator square root of 3 end fraction end cell row cell fraction numerator 1 over denominator cos space theta end fraction end cell equals cell fraction numerator 2 over denominator square root of 3 end fraction end cell end table

Solve the resulting equation to find theta

cos space theta equals fraction numerator square root of 3 over denominator 2 end fraction

theta equals pi over 6

(c) Find the average distance from the origin to a point on the curve.

The average distance from the origin is given by the formula fraction numerator 1 over denominator beta minus alpha end fraction integral subscript alpha superscript beta r open parentheses theta close parentheses space d theta

Substitute alpha equals 0, beta equals pi over 4 and r open parentheses theta close parentheses equals sec squared space theta into the formula and integrate

fraction numerator 1 over denominator pi over 4 minus 0 end fraction integral subscript 0 superscript pi over 4 end superscript sec squared space theta space d theta
equals 4 over pi open square brackets tan space theta close square brackets subscript 0 superscript pi over 4 end superscript
equals 4 over pi open parentheses tan space pi over 4 minus tan space 0 close parentheses
equals 4 over pi open parentheses 1 minus 0 close parentheses

4 over pi

(d) Find the equation of the curve in terms of x and y. Give your answer in the form y equals f open parentheses x close parentheses.

One way is to start by multiplying both sides of the polar curve by sec squared space theta

table row r equals cell sec squared space theta end cell row cell r space cos squared space theta end cell equals 1 end table

The left-hand side is almost the square of x equals r space cos space theta, so multiply both sides by r

r squared space cos squared space theta equals r

Now substitute x equals r space cos space theta into the left-hand side and r equals square root of x squared plus y squared end root into the right-hand side, to give an equation just in x and y

x squared equals square root of x squared plus y squared end root

The question wants the answer in the form y equals f open parentheses x close parentheses so make y the subject

table row cell x to the power of 4 end cell equals cell x squared plus y squared end cell row cell x to the power of 4 minus x squared end cell equals cell y squared end cell row cell plus-or-minus square root of x to the power of 4 minus x squared end root end cell equals y end table

To decide which sign to choose, first look at the sketch of the curve given in the question

The curve is in the first quadrant, meaning y greater or equal than 0, so chose the positive square root

y equals square root of x to the power of 4 minus x squared end root

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Mark Curtis

Author: Mark Curtis

Expertise: Maths

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.

Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.