Cartesian to Polar Coordinates Calculator

$\color{red} \bold{Related \space Calculators:}$
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Cylindrical to Cartesian coordinates
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Spherical to Cartesian coordinates
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$\bold{Table \space of \space Content}$

• 1. Introduction to the Cartesian to Polar coordinates calculator
• 2. What is the Formulae used ?
• 3. How do I convert the Cartesian coordinates to Polar?
• 4. Why choose our Cartesian to Polar coordinates Calculator?
• 5. A Video for explaining this concept
• 6. How to use this calculator ?
• 7. Solved Examples
• 8. Frequently Asked Questions (FAQs)
• 9. What are the real-life applications?
• 10. Conclusion

1. Introduction to the Cartesian to Polar coordinates calculator

Embarking on a journey through the mathematical landscape, we delve into the conversion from Cartesian to polar coordinates. Join us as we unravel the essence of this transformation, exploring its definition, practical applications, and the step-by-step process to navigate between two coordinate systems seamlessly.
$\bold{What \space are \space Cartesian \space Coordinates?}$
Cartesian coordinates locate points in the argand plane using a pair of perpendicular axes (usually denoted as x and y) intersecting at the origin (0, 0). A Point is represented as (x, y), where x represents the x coordinate or horizontal distance and y represents the y coordinate or vertical distance of the point from the origin. We also call it a rectangular coordinate system. Here, x and y can attain any real value.
$\bold{What \space are \space Polar \space Coordinates?}$
Polar coordinates, on the other hand, use a radial distance (r) from the origin and an angular measurement (θ) to locate points. A Point is represented as (r, θ), where r represents the distance of the end from the origin, and θ represents the angle (in degrees or radians) made by the line joining the origin to the point measured CCW from the positive x-axis. Due to mathematical convention, the value of the polar radius is taken as positive, and the range for the polar angle (θ) is [0, 2π].

2. What is the Formulae used?

$\bold{Radial \space distance}$ = r = $\sqrt{x^2 + y^2}$
To calculate the $\bold{Polar \space angle}$ θ first, we need to locate the point in the four quadrants in the xy-plane and then use the formula accordingly by considering all the cases.
Case 1: If x > 0 and y > 0 (Point lies in 1st quadrant) then, θ = $tan^{-1}(\frac{y}{x})$
Case 2: If x < 0 and y > 0 (Point lies in 2nd quadrant) then, θ = $\pi + tan^{-1}(\frac{y} {x})$
Case 3: If x < 0 and y < 0 (Point lies in 3rd quadrant) then, θ = $\pi + tan^{-1}(\frac{y}{x})$
Case 4: If x > 0 and y < 0 (Point lies in 4th quadrant) then, θ = $2\pi + tan^{-1}(\frac{y}{x})$
Case 5: If x = 0 and y > 0 (Point lies on positive y-axis) then, θ = $\frac{\pi}{2}$ radians or 90 degrees.
Case 6: If x = 0 and y < 0 (Point lies on negative y-axis) then, θ = $\frac{3\pi}{2}$ radians or 270 degrees.
Case 7: If x > 0 and y = 0 (Point lies on the positive x-axis), then θ = 0 radians or 0 degrees.
Case 8: If x < 0 and y = 0 (Point lies on the negative x-axis), then θ = $\pi$ radians or 180 degrees.
Case 9: If x = 0 and y = 0 (Point lies on Origin), then θ = Not defined.

3. How do I convert the Cartesian coordinates to Polar?

Identify the given cartesian coordinates (x, y) of the point.
Calculate the radial distance or magnitude of the point from the origin by using the formula r = $\sqrt{x^2 + y^2}.$
Determine the quadrant in which the point is lying.
Now determine the angle θ by using tan inverse formula by considering the cases written above.
Convert the angle to the desired units (degrees or radians), as the question asks. Write down the point's polar coordinates (r, θ).

4. Why choose our Cartesian to Polar coordinates Calculator?

$\bold{Easy \space \space to \space Use}$
Our calculator page provides a user-friendly interface that makes it accessible to both students and professionals. You can quickly input your square matrix and obtain the matrix of minors within a fraction of a second.

$\bold{Time \space Saving \space By \space automation}$
Our calculator saves you valuable time and effort. You no longer need to manually calculate each cofactor, making complex matrix operations more efficient.

$\bold{Accuracy \space and \space Precision}$
Our calculator ensures accurate results by performing calculations based on established mathematical formulas and algorithms. It eliminates the possibility of human error associated with manual calculations.

$\bold{Versatility}$
Our calculator can handle all input values like integers, fractions, or any real number.

$\bold{Complementary \space Resources}$
Alongside this calculator, our website offers additional calculators related to Pre-algebra, Algebra, Precalculus, Calculus, Coordinate geometry, Linear algebra, Chemistry, Physics, and various algebraic operations. These calculators can further enhance your understanding and proficiency.

5. A video based on how to convert the Cartesian coordinates to Polar.

6. How to use this calculator

This calculator will help you to convert the cartesian coordinates to polar.
In the input boxes, you must put the values x and y.
After clicking the Calculate button, a step-by-step solution will be displayed on the screen.
You can access, download, and share the solution.

7. Solved Examples

$\bold{Question: 1}$
Convert the given Cartesian coordinates as (1, $\sqrt{3}$) into polar coordinates.
$\bold{Solution:1}$
r = $\sqrt{(1)^2 + (\sqrt{3})^2}$ = 2
Since x > 0 & y > 0, the point lies in the 1st quadrant.
Polar angle = $\theta = tan^{-1}(\frac{\sqrt{3}}{1})$ = 60 degrees or $\frac{\pi}{3}$ radian

$\bold{Question: 2}$
Convert the given Cartesian coordinates as (-1, $\sqrt{3}$) into polar coordinates.
$\bold{Solution:2}$
r = $\sqrt{(-1)^2 + (\sqrt{3})^2}$ = 2
Since x < 0 & y > 0, the point lies in the 2nd quadrant.
Polar angle = $\theta = \pi + tan^{-1}(\frac{\sqrt{3}}{-1})$ = 120 degrees or $\frac{2\pi}{3}$ radian

$\bold{Question: 3}$
Convert the given Cartesian coordinates as (-1, -$\sqrt{3}$) into polar coordinates.
$\bold{Solution:3}$
r = $\sqrt{(-1)^2 + (-\sqrt{3})^2}$ = 2
Since x < 0 & y < 0, the point lies in the 3rd quadrant.
Polar angle = $\theta = \pi + tan^{-1}(\frac{-\sqrt{3}}{-1})$ = 240 degrees or $\frac{4\pi}{3}$ radian

$\bold{Question: 4}$
Convert the given Cartesian coordinates as (1, -$\sqrt{3}$) into polar coordinates.
$\bold{Solution:4}$
r = $\sqrt{(1)^2 + (-\sqrt{3})^2}$ = 2
Since x > 0 & y < 0, the point lies in the 4th quadrant.
Polar angle = $\theta = 2\pi + tan^{-1}(\frac{-\sqrt{3}}{1})$ = 300 degrees or $\frac{7\pi}{3}$ radian

8. Frequently Asked Questions (FAQs):-

Can polar coordinates represent any point in the Cartesian plane?

Yes, polar coordinates can represent any point, and the conversion is unique.

What does the angle θ represent in polar coordinates?

The angle θ represents the direction of the point from the positive x-axis.

Can negative values be present in polar coordinates?

The radius (r) is always non-negative, but the angle θ can take negative values.

How does the conversion handle points on the axes?

For points on the axes, θ may be undefined or take special values (e.g. θ = 0 on the positive x-axis).

Is there a difference between polar and Cartesian distance?

No, the polar distance (r) is equivalent to the Cartesian distance $\sqrt{x^2 + y^2}$

9. What are the real-life applications?

In navigation, converting Cartesian to polar coordinates finds application in determining the distance and direction of a ship or aircraft from a reference point. This aids in effective route planning and location tracking.

10. Conclusion

As we conclude our exploration of the conversion of Cartesian to polar coordinates, we recognize its role as a versatile tool for expressing points with distance and direction. Embrace the simplicity of the formulas and their significance in applications ranging from mathematics to navigation. Though rooted in basic trigonometry, this transformation stands as a fundamental bridge, seamlessly connecting the rectangular and polar realms in the vast landscapes of mathematical coordinates.

This blog is written by Neetesh Kumar

If you have any suggestions regarding the improvement of the content of this page, please write to me at My Official Email Address: [email protected]

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