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Rectangular to Polar Equation Calculator + Online Solver With Free Steps

The Rectangular to Polar Equation calculator deals with two coordinate systems: the rectangular or the Cartesian Coordinate System and the Polar Coordinate System.

These two systems are used to determine the position of a point in a 2D plane. The Rectangular to Polar Equation calculator is used to determine the position of the point $P(x,y)$ by finding the polar coordinates ($r$,$θ$).

What Is a Rectangular to Polar Equation Calculator?

A rectangular to polar equation calculator is an online calculator that converts two-dimensional rectangular coordinates into polar coordinates.

This calculator takes rectangular components $x$ and $y$ as input where $x$ is the distance of a point P from the origin (0,0) along the $x$-axis and $y$ is the distance of the point $P$ from the origin along the $y$-axis.

The polar coordinates $r$ and $θ$ give the position of point P where $r$ is the radius of the circle or the distance traveled from the center of the circle to the point $P$. $θ$ is the angle from the positive $x$-axis in the counterclockwise direction.

Polar equation is given as:

\[ y = r (e)^{ι.θ} \]

It is obtained from the rectangular coordinate equation $(x+ιy)$.

How To Use Rectangular to Polar Equation Calculator

Here are the steps required to use the rectangular to polar equation calculator.

Step 1:

Enter the $x$ and $y$ coordinate values against the blocks titled x and y respectively.

Step 2:

Press the submit button for the calculator to process the polar coordinates $r$ and $θ$.

Output:

The output will show four windows as follows:

Input Interpretation:

The calculator shows the interpreted values for the $x$ and $y$ coordinates for which the polar coordinates are determined. The default values set for the $x$ and $y$ coordinates are 3 and -2, respectively.

Result:

The result block shows the values for $r$ and $θ$. The value of $r$ is obtained by putting the values of $x$ and $y$ in the following equation:

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

The value of $r$ shows the vector length or magnitude of the resultant vector which is always a positive value.

Also, the value of $θ$ is obtained by putting the values of $x$ and $y$ in the following equation:

\[ \theta = \arctan (\frac{y}{x}) \]

The positive value of $θ$ shows a counter-clockwise direction from the $x$-axis and the negative value shows a clockwise direction from the $x$-axis.

Vector Plot:

The vector plot shows a 2D graph with positive and negative $x$ and $y$ rectangular coordinate axes.

The resultant vector is drawn by the output polar vectors ($r$, $θ$) with magnitude $r$ taken from the origin and angle $θ$ taken from the positive $x$-axis. The quadrant of the resultant vector is determined by the ($x$,$y$) coordinates displayed on the plot.

Vector Length:

The vector length shows the magnitude $r$ of the resultant vector.

Examples

Here are some examples that are solved using a Rectangular to Polar Equation Calculator.

Example 1:

For the rectangular coordinates

\[ (2, 2(\sqrt{3})) \]

find the polar coordinates (r,θ).

Solution:

\[ x = 2 \] and \[ y = 2(\sqrt{3}) \]

Putting the values of $x$ and $y$ in the equations of $r$ and $θ$:

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

\[ r = \sqrt{ (2)^2 + (2(\sqrt{3}))^2 } \]

\[ r = \sqrt{ 4 + 12 } \]

\[ r = \sqrt{ 16 } \]

\[ r = 4 \]

\[ \theta = \arctan (\frac{y}{x}) \]

\[ \theta = \arctan (\frac{2(\sqrt{3})}{2}) \]

\[ \theta = \arctan ( \sqrt{3} ) \]

\[ \theta = 60° \]

Figure 1 shows the resultant vector of example 1.

Figure 1

The same results are obtained using the calculator.

Example 2:

For the rectangular coordinates

\[ (-3(\sqrt{3}) , 3) \]

find the polar coordinates (r,θ).

Solution:

\[ x = -3(\sqrt{3}) \] and \[ y = 3 \]

Putting the values of $x$ and $y$ in the equation of $r$:

\[ r = \sqrt{ ( -3(\sqrt{3}) )^2 + ( 3 )^2 } \]

\[ r = \sqrt{ 27 + 9 } \]

\[ r = \sqrt{ 36 } \]

\[ r = 6 \]

For the value of θ, ignoring the negative sign of 3(\sqrt{3}) for the reference angle Φ.

The result is shown as:

\[ \Phi= \arctan (\frac{3} {3(\sqrt{3}) }) \]

\[ \Phi = \arctan (\frac{1} {\sqrt{3}}) \]

\[ \Phi = -30°  \]

Adding 180° to Φ will give the angle θ.

The angle θ is given as:

\[ \theta = -30° + 180° \]

\[ \theta = 150° \]

Figure 2 shows the resultant vector for example 2.

Figure 2

The same results are obtained using the calculator.

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All the images are created using GeoGebra.

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