What is Polar equations: Definition and 25 Discussions

In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian coordinate system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. The radial coordinate is often denoted by r or ρ, and the angular coordinate by φ, θ, or t. Angles in polar notation are generally expressed in either degrees or radians (2π rad being equal to 360°).
Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-17th century, though the actual term polar coordinates has been attributed to Gregorio Fontana in the 18th century. The initial motivation for the introduction of the polar system was the study of circular and orbital motion.
Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as spirals. Planar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more intuitive to model using polar coordinates.
The polar coordinate system is extended to three dimensions in two ways: the cylindrical and spherical coordinate systems.

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  1. T

    Integration of acceleration in polar coordinates

    I made this exercise up to acquire more skill with polar coordinates. The idea is you're given the acceleration vector and have to find the position vector corresponding to it, working in reverse of the image. My attempts are the following, I proceed using 3 "independent" methods just as you...
  2. F

    I Plotting polar equations and scale invariance

    Hello, In the plane, using Cartesian coordinates ##x## and ##y##, an equation (or a function) is a relationship between the ##x## and ##y## variables expressed as ##y=f(x)##. The variable ##y## is usually the dependent variable while ##x## is the independent variable. The polar coordinates...
  3. Saracen Rue

    I How to find the maximum arc length of this equation?

    After seeing a discussion about graphs of the relationship ##x^x + y^y = r^r##, it got me interested in attempting to see what the graphical appearance of ##{^{\infty}x}+{^{\infty}y}={^{\infty}r}## would look like. The first step I did was use the relationship of...
  4. Zack K

    Area between 2 polar equations

    Homework Statement Find the area of the region that lies inside the first curve and outside the second curve. ##r=6## ##r=6-6sin(\theta)## Homework Equations ##A=\frac {1} {2}r^2\theta## The Attempt at a Solution \[/B] If I'm correct, the area should just be ##\frac {1} {2}\int_{0}^{2\pi} 6^2...
  5. Vital

    Is there an algorithms to determine correct angles

    Homework Statement Hello! I am struggling with plotting polar graphs manually (without any help of the calculator). My main unresolved issue is with finding correct values of theta in a given range. For example, I have an equation: $r = cos(5\theta)$ Homework Equations and I know that the...
  6. Vital

    How is the xy curve formed from the rθ curve?

    Homework Statement Hello! I will be grateful for your help in deciphering the meaning of a paragraph from the book. I honestly don't understand how they got the semi-circle on the xy graph by transferring it from rθ graph. Homework Equations I attach the screen shot from the book. The Attempt...
  7. doktorwho

    Solving for the trajectory in the polar coordinate system

    Homework Statement On the surface of a river at ##t=0## there is a boat 1 (point ##F_0##) at a distance ##r_0## from the point ##O## (the coordinate beginning) which is on the right side of the coast (picture uploaded below). A line ##OF_0## makes an angle ##θ_0=10°## with the ##x-axis## whose...
  8. V

    Polar Coordinates: Position of Particle at T/8

    1. The question The position of a particle is given by r(t) = acos(wt) i + bsin(wt) j. Assume a and b are both positive and a > b. The plane polar coordinates of a particle at a time t equal to 1/8 of the time period T will be given by _ Homework Equations r(t) = acos(wt) i + bsin(wt) j. The...
  9. karush

    MHB Using desmos for Polar equations and the table feature

    Just couldn't find help with this anywhere but Using the desmos graphing calculator using the table feature I wanted to plot the points every $\frac{\pi}{12}$ $0<\theta<2\pi$ For $r=-2-3\sin(\theta)$ On a polar coordinate graph
  10. S

    Understanding the Polar-Cartesian Relationship in Jacobian Calculus

    This is a problem that has been bugging me all day. While working with the well-known dydx = rdrdθ, where r is a function of θ I divided both sides of the equation by dxdθ to get dy/dθ = r(dr/dx) For the left side, I use y = rsinθ and derive with respect to θ to get dy/dθ = sinθdr/dθ + rcosθ...
  11. S

    How do I finish this polar equations problem?

    Homework Statement Find the points on the given curve where the tangent is horizontal or vertical Homework Equations r = 3cos(θ) The Attempt at a Solution d/dθ = -3sin(θ) for horizontal: -3sin(θ)sin(θ) + 3cos(θ)cos(θ) I used identity and got: 3cos(2θ) = 0 I got the...
  12. Dethrone

    MHB Points of intersection of polar equations

    Find the points of intersection of $\rho=\cos\left({2\theta}\right)$ and $\rho=\cos\left({\theta}\right)$ By setting $\cos\left({2\theta}\right)=\cos\left({\theta}\right)$, we get the solutions $\theta=0,\frac{2\pi}{3},\frac{4\pi}{3}$. My question is how come that doesn't give us all the...
  13. S

    Finding the shared area of 2 polar equations

    Homework Statement Given the two polar equations r=5-3cos(θ) and r=5-2sin(θ) find the area of the region common to both curves. Homework Equations A= 1/2∫ r^2 dθ The Attempt at a Solution i understand that i plug in the two equations into the equation, but i don't know how to find the...
  14. A

    Finding the Area Inside Polar Equations

    1. Find the area of the region described: a) inside one loop of the lemniscate r^2=4cos(2theta) b) inside the six-petaled rose r^2=2sin(3theta) [b]2. A=integral [1/2 r^2 dtheta] Are there any easy ways to determine the integration bounds? (without graphing) Our textbook doesn't give any...
  15. J

    Double integrals of polar equations help

    Hey, everyone. I'm new to the forums and am hoping that someone can help me with a tricky problem from my multivariable calculus class. We're covering double integrals in polar coordinate systems, and there's a problem from my problem set for homework that I can't seem to get the grasp of. The...
  16. N

    Points of intersection with polar equations

    Homework Statement I have to find all of the points of intersection of the curves... r2 = sin(2θ) r2 = cos(2θ) The Attempt at a Solution sin(2θ) = cos(2θ) 2sinθcosθ = cos2θ - sin2θ 2sinθcosθ - cos2θ = -sin2θ cosθ(2sinθ - cosθ) = -sin2θ This is where I'm having a problem, I'm...
  17. N

    Intersection pts of polar equations

    Homework Statement I have to find the area of the region that lies inside the curves: r = sin(θ) r = sin(2θ) The Attempt at a Solution I'm assuming the first step would be to find the points of intersection so I know WHERE to integrate from/to, so I set the equations equal to each...
  18. R

    Finding Directrix and Sketching Hyperbola in Polar Coordinates

    Homework Statement Hyperbola formula 9x^2 - 4y^2 + 36x + 24y - 36 = 0. Convert to rectangular form, find coordinates of the vertices, find coordinates of the foci, find eccentricity, what is the equation of the conic section in polar coordinates if the pole is taken to be the leftmost focus...
  19. D

    Polar equations of conics question

    for this problem, I've been given the vertices of the hyperbola as (4, pi/2) and (-1, 3pi/2). the question asks to find the polar equation of this hyperbola. so what i did was do a quick sketch of the graph. (4, pi/2) is essentially (0,4) and (-1, 3pi/2) is essentially (0,1). the midpoint of...
  20. S

    Changing polar equations to rectangular equations?

    changing polar equations to rectangular equations? Can somebody please explain to me, how I would convert: ?=?/2 into a rectangular equation? Along with: r=sin?, r=6cos+sin?, r(squared)sin2?=2 Your help would be greatly appreciated!
  21. J

    Intersection Points of Polar Equations

    I've been having a problem finding the intersection points of the following polar equations. r=1+3sin(theta) and r=1-3cos(theta) Now I've set the equations equal to each other to obtain those points. I've set each equation equal to zero. The problem I'm having is that when graphed...
  22. J

    How to Find the Area of a Polar Equation Excluding Overlapping Part?

    It's my first time here so I guess I have to introduce myself first. I am a junior in high school taking Calculus BC and Physics B (taking Physics C next year). Currently, as of September 29, 2008, my class is learning of Polar equations. We just went over basics but I am not perfectly...
  23. C

    Convertong polar equations to rectangular eq.

    Homework Statement (theta)= 45Degrees Homework Equations (theta)= Arctan(y/x) (x>0) (theta)= Arctan(y/x) (x<0) The Attempt at a Solution Tan(45)=y/x 1 = y/x Does that mean Y = X? :confused:
  24. A

    TI-83 Troubleshooting: Graphing Polar Equations in Degrees & Radians

    Anyone know enough about TI-83s to help me out? For some reason whenever I try to graph a polar equation on mine it gives me what the graph should look like for radians when degrees is the mode of choice and vice versa. For example, if I were to graph sin(theta) I get a nice pretty circle when...
  25. M

    Integrating Polar Equations: Online Resources

    Can anyone point me to some online resources on how to integrate polar equations? Thanks.