Unit 10: Polar Coordinates and Complex Numbers
Locating points by distance and angle instead of x and y, polar curves that draw themselves as theta sweeps, and complex numbers plotted in the plane.
10 topics · 37 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
The Polar Coordinate System
- How it worksHow do you plot a point given in polar coordinates like (3, 2pi/3) on a polar grid?
- Worked exampleFind three different polar coordinate pairs, including one with a negative r, that all name the point (2, π/4).
- Common mistakeWhy can the exact same point in polar coordinates be written in infinitely many different ways?
Your turn1 of 2
The point (7, 4pi/9) in polar coordinates can also be written as (-7, theta) for some angle theta between 0 and 2pi. Find theta.Converting Between Polar and Rectangular Coordinates
- How it worksHow do x = r·cos(theta) and y = r·sin(theta) convert a polar point into rectangular coordinates?
- Worked exampleFind the rectangular coordinates of the polar point (4, π/3).
- Worked exampleFind polar coordinates for the rectangular point (-3, 3).
- Common mistakeWhy can't you find theta from tan(theta) = y/x without also checking which quadrant the point is in?
Your turn1 of 2
Write the rectangular x-coordinate as a function of theta (in radians), given a fixed r = 11.Polar Graphs of Lines and Circles
- How it worksHow does holding r constant while theta sweeps a full circle trace out a polar circle centered at the pole?
- Worked exampleFind the rectangular equation equivalent to the polar equation r = 4·cos(theta).
- Common mistakeWhy doesn't theta = π/4 graph a single point the way r = constant graphs a circle?
- Ties togetherHow does a constant circle r = a·sin(theta) compare to the multi-petaled curves coming up next?
Your turn1 of 2
Find r for the polar equation r = 7·cos(theta) at theta = 3pi/9.Polar Roses
- How it worksHow does r = cos(n·theta) trace out one petal at a time as theta sweeps from 0 to π or 2pi?
- Worked exampleFind the number of petals drawn by r = sin(4·theta) and by r = sin(5·theta).
- Worked exampleFind the length of each petal of r = 3·sin(2·theta).
- Common mistakeWhy does an even n in r = cos(n·theta) produce twice as many petals as you might expect from n alone?
Your turn1 of 2
Find the length of each petal of r = 7·sin(5·theta).Cardioids and Limaçons
- How it worksHow does the ratio of the constant term to the coefficient in r = a + b·cos(theta) decide between a cardioid, a dimpled limaçon, or one with an inner loop?
- Worked exampleFind whether r = 2 + 3·sin(theta) produces a limaçon with an inner loop, and find where the loop crosses the pole.
- Common mistakeWhy isn't every curve of the form r = a + b·cos(theta) automatically a cardioid?
- Common mistakeWhy does a limaçon's inner loop cross the pole at two different angles instead of just one?
Your turn1 of 2
Find the ratio a/b for r = 4 + 5·sin(theta), which classifies the curve (less than 1 gives an inner loop).Lemniscates and Other Polar Curves
- How it worksHow does squaring r in a lemniscate's equation restrict which angles theta are even allowed?
- Worked exampleFind the values of theta for which r2 = 9·cos(2·theta) has no real solution for r.
- Common mistakeWhy can't theta take on every value from 0 to 2pi when graphing a lemniscate?
Your turn1 of 2
Find the smallest positive value of theta for which r2 = 7·cos(4·theta) stops having a real solution for r.Complex Numbers in the Complex Plane
- How it worksHow does plotting a + bi as the point (a,b) turn complex numbers into geometric objects?
- Worked exampleFind the location and distance from the origin of the complex number 3 - 4i in the complex plane.
- Common mistakeWhy isn't the imaginary part of a complex number itself an imaginary distance on the plane?
- Ties togetherHow does plotting a complex number as a point set up rewriting it with a distance and an angle instead?
Your turn1 of 2
Find the distance from the origin of the complex number x - 11i in the complex plane, as a function of x.Polar Form of a Complex Number
- How it worksHow do the modulus and argument of a complex number convert a + bi into r·(cos(theta) + i·sin(theta))?
- Worked exampleFind the polar form of the complex number -1 + i*√3.
- Common mistakeWhy isn't the modulus of a complex number the same as its real part?
Your turn1 of 2
Find the modulus of the complex number x + 9i, as a function of x.Multiplying and Dividing Complex Numbers in Polar Form
- How it worksHow do moduli multiply and arguments add when two complex numbers in polar form are multiplied?
- Worked exampleFind the product of 2·(cos(30 degrees)+i·sin(30 degrees)) and 3·(cos(60 degrees)+i·sin(60 degrees)).
- Common mistakeWhy doesn't multiplying complex numbers in polar form require expanding anything with FOIL?
- Ties togetherHow does multiplying a complex number by itself repeatedly in polar form lead toward DeMoivre's Theorem?
Your turn1 of 2
Two complex numbers in polar form have moduli 7 and r2. Write the modulus of their product as a function of r2.DeMoivre's Theorem and Roots of Complex Numbers
- How it worksHow does DeMoivre's Theorem turn raising to a power into multiplying the angle and raising the modulus?
- Worked exampleFind (1+i)6 using DeMoivre's Theorem.
- Worked exampleFind all three cube roots of the complex number 8.
- Common mistakeWhy doesn't a complex number have just one cube root the way a positive real number seems to?
Your turn1 of 2
Write the modulus of (2·(cos(42 degrees)+i·sin(42 degrees)))n using DeMoivre's Theorem, as a function of n.