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Understanding how to convert coordinates from polar to rectangular opens up a world of possibilities in mathematics, physics, and computer science. Polar coordinates, which use distance (r) and angle (*θ*), can be seamlessly transformed into rectangular coordinates (x, y) using trigonometric functions. This conversion allows for more intuitive analysis and representation of points in a two-dimensional plane.

Polar to Rectangular conversion is the process of translating a point specified in polar coordinates to its equivalent position in rectangular coordinates. It involves using trigonometric functions to calculate the horizontal (x) and vertical (y) distances from the origin.

The conversion is accomplished through two simple formulas:

x= r⋅cos(θ)

y= r⋅sin(θ)

Here's an explanation of each part of the formula:

**x=r⋅cos(θ):**- In this formula, r represents the radial distance from the origin to the point in polar coordinates. It is the length of the line segment connecting the origin (0,0) to the point in question
- (
*θ*) denotes the angle formed between the positive x-axis and the line segment connecting the origin to the point. This angle is measured in radians. - The convention of using "." for multiplication in mathematics, rather than "x," ensures clarity and avoids confusion with the variable "x."
- cos(
*θ*) is the cosine of the angle (*θ*), representing the ratio of the adjacent side (x-coordinate) to the hypotenuse radius (r).

Therefore, (x) is the horizontal distance from the origin to the point, calculated by multiplying the radial distance (r) by the cosine of the angle (

*θ*).**y= r⋅sin(***θ*):- Similar to the x-coordinate, y is the vertical distance from the origin to the point.
- The convention of using "." for multiplication in mathematics, rather than "x," ensures clarity and avoids confusion with the variable "x."
- sin(
*θ*) is the sine of the angle (*θ*), representing the ratio of the opposite side (y-coordinate) to the hypotenuse radius (r). - By multiplying the radial distance (r) by the sine of the angle (
*θ*), you obtain the vertical distance (y) from the origin to the point.

These formulas essentially describe how to decompose a polar coordinate (r, *θ*) into its rectangular components (x, y) based on trigonometric functions. The cosine function is used for the x-coordinate (horizontal position), and the sine function is used for the y-coordinate (vertical position). Together, these formulas enable the conversion between different coordinate systems, allowing us to represent points in both polar and rectangular coordinates.

Example 1:

$Polar\; Coordinates:(r=5,\theta =\frac{\pi}{4})$

Conversion:

$x=5.\mathrm{cos}\left(\frac{\pi}{4}\right)$

$y=5.\mathrm{sin}\left(\frac{\pi}{4}\right)$

Result:

Rectangular Coordinates (x,y) = ( $\frac{5}{\sqrt{2}}$, $\frac{5}{\sqrt{2}}$)

Example 2:

$Polar\; Coordinates:(r=3,\theta =\frac{\mathrm{3\pi}}{4})$

Conversion:

$x=3.\mathrm{cos}\left(\frac{\mathrm{3\pi}}{4}\right)$

$y=3.\mathrm{sin}\left(\frac{\mathrm{3\pi}}{4}\right)$

Result:

Rectangular Coordinates (x,y) = (0, -3)

Example 3:

$Polar\; Coordinates:(r=4,\theta =\frac{\pi}{3})$

Conversion:

$x=4.\mathrm{cos}\left(\frac{\pi}{3}\right)$

$y=4.\mathrm{sin}\left(\frac{\pi}{3}\right)$

Result:

Result: Rectangular Coordinates (x,y) = (2, $2\sqrt{3}$)

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