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Polar Coordinates Calculator

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An online polar coordinates calculator will display the conversion of polar to Cartesian coordinate and Cartesian to polar coordinates. This calculator provides the stepwise results for the 2-D space of 3-D coordinates. Let’s take a look at how to convert polar coordinates to rectangular coordinates and vice versa using their formulas.

What is the Polar Coordinate?

In mathematics, rectangular or Cartesian coordinates are a pair of coordinates that are measured along the axis. These are perpendicular to each other. Similarly, these paired coordinates defines a position in a plane. The first coordinate defines the length of the line that connects the point and the origin, and the second coordinate defines the angle formed by the line. Therefore, the polar to rectangular equation calculator compute the angle in radian.

Cartesian and Polar Coordinates:

Coordinates are used to uniquely describe the position of a point in space. This means that we only have two dimensions: height and width (no depth), just like on a piece of paper. Create a Cartesian coordinate system by drawing lines that are perpendicular to each other. Therefore, the point where they meet is called the origin. The coordinate of any point in space is the distance between that point and the two lines indicated by the x-axis and y-axis.

The origin of the polar system includes all vertical lines. The source of a polar system is a point called a pole. An arbitrary ray starting from this point will be selected as the polar axis. To find the polar coordinates of a given point, the rectangular to polar coordinates calculator must find and draw a connecting line first. Then, the coordinates of these points are the length of the line r and the angle θ between the polar axis. Our polar coordinates calculator can do the conversion for Cartesian and polar.

However, an online Linear Interpolation Calculator allows you to find the interpolated values for the data points on a line or a curve.

Polar to Rectangular Coordinate:

In Mathematics, polar to rectangular coordinates represents the conversion of polar to rectangular. The rectangular coordinates are called the Cartesian coordinate which is of the form (x, y), whereas the polar coordinate is in the form of (r, θ). The conversion formula is used by the polar to Cartesian equation calculator as:

$$ x = r cos θ $$

$$ y = r sin θ $$

Now, the polar to rectangular equation calculator substitute the value of r and θ in the conversion formula and solve for the x and y values to get the rectangular coordinates.

To do it, simply polar coordinate calculator use the following polar equation to rectangular:

$$ x = r * cos θ y = r * sin θ $$

The value y/x is the slope of the line that joining the pole and the arbitrary point.

Example:

Convert (r, θ) = (2, 9) to Cartesian coordinates.

Solution:

To convert this the polar to rectangular calculator use the formula:

$$ X = r * cos θ, y = r * sin θ $$

So, substitute the values

$$ x = 2 * cos (9), y = 2 * sin (9) $$

Therefore, the cartesian to polar calculator display answer for x and y coordinate.

$$ (x, y) = (-1.8222, 0.8242) $$

However, an online Parallel and Perpendicular Line Calculator will help you to determine the equation of the perpendicular and parallel to the given line that passes through the entered points.

Cartesian Coordinates to Polar Coordinates:

To represent the Cartesian coordinates as polar. Let’s assumes that the origin of the Cartesian coordinate system overlaps the poles of the polar coordinate system. So, rectangular to polar equation calculator use the following formulas for conversion:

$$ r =  \sqrt{(x^2 + y^2)} $$

$$ θ = arctan (y/x) $$

Where,

  • (x, y) rectangular coordinates;
  • (r, θ) polar coordinates.

The following restrictions by rectangular to polar calculator to convert the coordinates:

  • r must be greater than or equal to 0;
  • θ must be in the range of [-π, π].

Example:

Convert (r, θ) = (2, 6) to polar coordinates.

Solution:

Converting to coordinates to find r, we have r = \( \sqrt{x^2 + y^2}\)

So,

$$ r = \sqrt{2^2 + 6^2} = 6. 3245 $$

To find θ, polar to rectangular calculator use angle θ as:arctan (x / y)

θ = arctan (6 / 2) = 1.249

Answer:              

$$ (r, θ) = (6.3245, 1.249) $$

It is also possible that r is negative. In this case, polar to rectangular equation calculator add/subtract π from the found θ:

Then, (r, θ) = (- 6.3245, 4.3906)

How Polar Coordinates Calculator Works?

An online polar coordinate calculator will convert polar coordinates to rectangular and vice versa by following these instructions:

Input:

  • Firstly, select the conversion type from the drop-down list such as Cartesian to polar or polar to Cartesian.
  • Now, substitute the values in the related fields.
  • Hit the calculate button to see the conversion.

Output:

  • The Polar Coordinates Calculator provides the conversion according to the selection in degree and radian.
  • Polar to Cartesian equation calculator also displays step-by-step calculations of coordinates.

FAQ:

Which type of works uses polar coordinates?

Polar coordinates are used in aviation, animation, computing, engineering, architecture, and the military.

What is the difference between cylindrical coordinates and polar coordinates?

Cylindrical coordinates are a simple extension of 2D coordinates to three dimensions. These coordinates are the distance from the point to the origin. The coordinate θ is the angle between the x-axis coordinate and the line segment from the origin to the point.

What does the Cartesian coordinate system mean in physics?

The Cartesian coordinate system is defined in two dimensions (also called a Cartesian coordinate system or an orthogonal coordinate system). An ordered pair of vertical lines (axes), the length units of the two-axis, and the direction of each axis.

Conclusion:

Use this Polar Coordinates Calculator for the transformation of polar and rectangular coordinates. A 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.

Reference: 

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.