If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Multivariable calculus

### Course: Multivariable calculus>Unit 4

Lesson 6: Double integrals (articles)

# Polar coordinates

Introduction to polar coordinates.

## Want to join the conversation?

• What's the advantage of using Polar coordinates over Cartesian coordinates ?
• Some integrals are easier to solve in polar coordinates rather than cylindrical coordinates; in polar coordinates a rectangle is an annulus/circle in cartesian coordinates. So problems involving circles can be simplified by switching over to polar coordinates. There are countless examples of this type of reasoning - the problem is very hard in one coordinate system but becomes much simpler in another.
• Are there any practice problems for polar coordinates?
• will arctan(4/3) = θ always work? because from what I learned, arctan only gives u angles in the 1st & 4th quadrant (the range of arctan(o/a) = θ, θ can only be from -90 < θ < 90, exclusive) so what happens if the point is in the 2nd or 3rd quad? example: (-3,4) or (-3,-4). I once had a point in the 2nd quad (-1/2, √3/2), tan(120) = -√3, but arctan(-√3) does NOT give me 120 back, it gave me -60 instead
• tan(120)=tan(-60). The range of arctan is only between -90 and 90, so it returns -60 instead of 120.
• How to convert 3D Cartesian coordinates to 3D Polar Coordinates
• Sorry if I'm late.There is no 3D polar coordinates.However, There is spherical coordinates which are very similar to polar coordinates,but we use a third angle (phi) to measure angle between the straight line from the origin to the point and its z-axis projection. theta(the polar angle) will measure the angle between its xy-plane projection and the x-axis . To convert from 3D Cartesian coordinate to spherical coordinates , we use the following formulas:
r = sqrt(x^2+y^2+z^2) , theta(the polar angle) = arctan(y/x) ,
phi (the projection angle) = arccos(z/r)
edit: there is also cylindrical coordinates which uses polar coordinates in place of the xy-plane and still uses a very normal z-axis ,so you make the z=f(r,theta) in cylindrical cooridnates
• I am doing algebra II and I can't find any videos on how to convert decimal numbers into polar form. Can anyone explain how to convert decimal numbers into polar form?
• What's the difference between the polar coordinate and a vector in standard form? They seem like they're doing the same thing.
(1 vote)
• They are just two different ways to represent a point in a Cartesian Plane. An imaginary number in rectangular form could actually be written in vector standard form.
The polar coordinates can be helpful if we are more interesting in things like the rotation of the vector, which polar coordinates easily give.

Hope this helps,
- Convenient Colleague
• Can i see more questions for practice!
• At can Theta be negative? because tan can be negative.
• What does r represent in polar coordinate?
(1 vote)
• 'r' represents the distance from the origin. It can be interpreted as the radius of a circle centered at the origin with the selected point on the circumference. This is why any point (x,y) is represented as (rcos(theta), resin(theta)).
• What if, for example, the point is (0,5). How would it work in arctan since that would require y/x, which is 5/0, making it indivisible since 0 is in the denominator?
(1 vote)
• You are right, we can't use arctan to find the angle.

Instead we have to realize that (0, 5) represents the pure imaginary number 5𝑖

Since the coefficient (5) is positive we know that the angle is 𝜋∕2

If the coefficient is negative the angle is −𝜋∕2