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Make conversions between rectangular (cartesian) and polar plane coordinates.

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This calculator is used to convert rectangular to polar coordinates and vice versa for a 2D system. It shows the step-by-step calculation for both conversions.

Polar coordinates represent a point that is positioned with respect to the origin (0,0) and an angle from the origin in the reference direction. They are written as (r, θ), such that:

- r is the radial distance from the origin to point
- θ is the angle between the x-axis and the line joining point with the origin.

Cartesian coordinates indicate a point in an XY-plane by a pair of numerical values. These coordinates are signed distances from a point to two perpendicular directed lines. They are usually written as (x, y), such that:

- X represents the horizontal distance from the origin (positive to the right, negative to the left).
- Y represents the vertical distance from the origin (positive upwards, negative downwards

There are two methods of converting between the coordinates:

**2.1.1 To Convert from Cartesian to Polar**

For rectangular coordinates (x, y) given in the above picture, you can find polar coordinates (r, θ) as follows:

\(r = \sqrt{x^{2} + y^{2}}\)

\(θ = arctan (\dfrac{y}{x}\)

**2****.1.2 To Convert from Polar to Cartesian**

Use the following polar equations to cartesian equations for converting: \(x = r cos θ\) \(y = r sin θ\)

- Select the conversion type
- Enter the required coordinate values
- Click ‘Calculate’ and get converted form

**Solution:**

**Data Given:**

- Vertical coordinate = y = 2
- Horizontal coordinate = x = 6

**Calculations:**

**Step 1: Determine ‘r’**

\(r = \sqrt{x^{2} + y^{2}}\)

\(r = \sqrt{6^{2} + 2^{2}}\)

\(r = \sqrt{36 + 4}\)

\(r = \sqrt{40}\)

\(r = 6.324555320336759\)

**Step 1: Determine ‘θ’**

\(θ = arctan (\dfrac{y}{x}\)

\(θ = arctan (\dfrac{2}{6}\)

\(θ = arctan (\dfrac{1}{3}\)

**Solution:**

**Data Given:**

- r = 2
- \(θ = 45^\text{o}\)

**Calculations:**

**Step 1: **Determine Horizontal Rectangular Component **‘x’**

\(x = r cos θ\)

\(x = 2 * 0.707\)

\(x = 1.414\)

**Step 2:** Determine Vertical Rectangular Component **‘y’**

\(y = r sin θ\)

\(y = 2 * sin\left(45^\text{o}\right)\)

No, the polar coordinates of a point are not unique. The reason is that each point can be represented by infinite polar coordinates in infinite ways.

In polar coordinates, z represents the complex number in polar form, such that: z = x + iy (where i (iota) = \(\sqrt{-1}\)

**Where**

- x = Real part
- y = Imaginary part

The polar coordinate representation of 1 is given as:

\(1 = 1e^{i} 0\)

The same equation holds true for \(1 = 1e^{i} 2π\) due to the fact that sine and cosine are both periodic with the angle 2π.

Cartesian Coordinates (x, y) | Polar Coordinates (r, θ) |
---|---|

(3, 4) | √(25) , arctan(4/3) ≈ 53.13° |

(-2, 1) | √(5) , arctan(-1/2) ≈ -26.57° |

(0, 5) | 5, 90° |

(-4, 4) | √(32) , arctan(1/1) ≈ 45° (or 315°) |

This polar coordinates calculator can handle all the conversions between coordinates including all ones listed in the above table, with the complete solution shown.

From the source of Wikipedia: Polar coordinate system, Conventions, Uniqueness of polar coordinates, Converting between polar and Cartesian coordinates.

From the source of Math Insight: Conversion formulas, the plane of Cartesian coordinates, coordinates r and θ.

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