# Cartesian to polar confusion (simple)?

• noahsdev
In summary, the complex number lies in quadrant 1, the tan function is incorrect, and the angle is found using cosine and secant.
noahsdev

## Homework Statement

Convert -2+2√3i to polar coordinates.

r = √x2+y2
θ = tan-1(y/x)

## The Attempt at a Solution

I am confused because θ = tan-1(2√3/2) = tan-1(√3) = -π/3 and r = 4, so that would make the polar form 4cis(-π/3), but the calculator gives: 4cis(2π/3).
I think the calculator is right because when I convert my answer (4cis(-π/3)) back to cartesian it gives -2-2√3i, whereas the other (4cis(2π/3))gives the right answer, -2+2√3i.

Can someone explain what I'm doing wrong?
Thanks. :)

In which quadrant does your complex number lie?
For which interval of angles is the standard tangent function defined?

Tangent is, of course, periodic and your calculator can give only one value- the "principal" value which, for tangent, is the value of $\theta$ with the smallest absolute value. Since tangent is periodic with period $\pi$, $tan(-\pi/3)= tan(-\pi/3+ \pi)= tan(2\pi/3)$.

You distinguish between them by noting that $-\pi/3$ is in the fourth quadrant, (+,-), while $2\pi/3$ is in the second quadrant, (-, +).

arildno said:
In which quadrant does your complex number lie?
For which interval of angles is the standard tangent function defined?
OK, I have found the angle using x and y (cos and sin) and they both confirm that the calculator is correct. And yes, it does make sense since the complex number lies in quadrant 1 but why is the tan function wrong? I'm guessing you were hinting at that part but I really don't know. :)

noahsdev said:
OK, I have found the angle using x and y (cos and sin) and they both confirm that the calculator is correct. And yes, it does make sense since the complex number lies in quadrant 1 but why is the tan function wrong? I'm guessing you were hinting at that part but I really don't know. :)

Are you sure that -2+2SQRT(3)i is in the first quadrant? Why don't you make a sketch?

HallsofIvy said:
Tangent is, of course, periodic and your calculator can give only one value- the "principal" value which, for tangent, is the value of $\theta$ with the smallest absolute value. Since tangent is periodic with period $\pi$, $tan(-\pi/3)= tan(-\pi/3+ \pi)= tan(2\pi/3)$.

You distinguish between them by noting that $-\pi/3$ is in the fourth quadrant, (+,-), while $2\pi/3$ is in the second quadrant, (-, +).
Yes that makes sense. Thanks.
P.S I know the quadrants haha I misstyped :)

## 1. How do I convert Cartesian coordinates to polar coordinates?

To convert Cartesian coordinates (x,y) to polar coordinates (r,θ), use the following formulas:
r = √(x² + y²)
θ = tan⁻¹(y/x)

## 2. What is the difference between Cartesian and polar coordinates?

Cartesian coordinates use x and y values to represent a point on a plane, while polar coordinates use r and θ values to represent a point in terms of distance and angle from the origin.

## 3. Can I convert polar coordinates to Cartesian coordinates?

Yes, you can convert polar coordinates to Cartesian coordinates using the formulas:
x = r*cos(θ)
y = r*sin(θ)

## 4. How do I plot points in polar coordinates?

To plot a point in polar coordinates, first identify the value of r (distance from the origin) and θ (angle from the positive x-axis). Then, starting from the origin, move r units in the direction of θ and mark the point.

## 5. What is the purpose of using polar coordinates instead of Cartesian coordinates?

Polar coordinates are useful for representing points in terms of distance and angle, which can be more intuitive in certain situations such as graphing circles or other curved shapes. They can also simplify certain mathematical operations, such as calculating the distance between two points.

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