Cartesian to polar conversions

In summary: For example, find the length of the arc of a circle of radius 2 meters subtended by a central angle of 30 degrees. In that case, you would need to convert the angle to radians: the length of the arc is (1/6) of the circumference which is 2\pi meters. The central angle is 30 degrees so the subtended arc is (30/360)(2\pi)= \pi/6 meters.)In summary, to find the polar coordinates of a point given its cartesian coordinates, you can use the equations r^2 = sqrt(a^2 + b^2) and tan(theta) = b/a. In the given example of the point (3sqrt(
  • #1
rcmango
234
0

Homework Statement



find polar coordinates of the points whose cartesian coordinates are given.

Homework Equations



heres the point: (3sqrt(3), 3)

The Attempt at a Solution



well i know that r^2 = (sqrt(a^2 + b^2))
so the answer here is : 6

and if we use tan(theta) = o/a = 3/(3sqrt(3)
so the answer here is: 1/sqrt(3)

so theta is pi/6

so how do i know its pi/6?
how do i convert to get this answer with pi? radians?

help please.
 
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  • #2
theta = arctan(1/sqrt3) = pi/6 (in radians, always radions for polar coordinates).
 
  • #3
rcmango said:

Homework Statement



find polar coordinates of the points whose cartesian coordinates are given.

Homework Equations



heres the point: (3sqrt(3), 3)

The Attempt at a Solution



well i know that r^2 = (sqrt(a^2 + b^2))
so the answer here is : 6

and if we use tan(theta) = o/a = 3/(3sqrt(3)
so the answer here is: 1/sqrt(3)
The right hand side of that equation, not the "answer" (to what question?!), is [itex]\frac{1}{\sqrt{3}}[/itex]. In general, if you know what [itex]tan(\theta)[/itex] is, you can find [itex]\theta[/itex] by using the arctan function- perhaps on a calculator. Here, you are probably expected to know that [itex]sin(\pi/6)= \frac{1}{2}[/itex] and that [itex]cos(\pi/6)= \frac{\sqrt{3}}{2}[/itex] so that [itex]tan(\pi/6)= \frac{1}{\sqrt{3}}[/itex].

so theta is pi/6

so how do i know its pi/6?
how do i convert to get this answer with pi? radians?

help please.

As benorin said- in polar coordinates the angle is always in radians. As a rule, the only time you use degrees is when the problem specifically involves angle that are given in degrees.
 

1. What is a Cartesian to polar conversion?

A Cartesian to polar conversion is a mathematical process used to convert coordinates from the rectangular coordinate system (Cartesian coordinates) to the polar coordinate system. This allows for the representation of a point in terms of its distance from the origin and its angle from the positive x-axis.

2. Why are Cartesian to polar conversions useful?

Cartesian to polar conversions are useful because they allow for a more intuitive way of representing coordinates that are based on distance and angle, rather than just x and y values. This can be particularly helpful in applications such as navigation, physics, and engineering.

3. How do you convert from Cartesian to polar coordinates?

To convert from Cartesian to polar coordinates, you can use the following formulas:

r = √(x2 + y2)

θ = tan-1(y/x)

Where r represents the distance from the origin and θ represents the angle from the positive x-axis.

4. Can you convert from polar to Cartesian coordinates?

Yes, you can also convert from polar to Cartesian coordinates. The formulas for this conversion are:

x = r * cos(θ)

y = r * sin(θ)

Where x and y represent the Cartesian coordinates and r and θ represent the polar coordinates.

5. Are there any limitations to using Cartesian to polar conversions?

One limitation of using Cartesian to polar conversions is that they are not always unique. This means that there can be multiple polar representations for the same point in the Cartesian system. Additionally, polar coordinates may not always be the most convenient representation for certain applications, so it is important to consider the context in which they are being used.

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