Transformation from rectangular to polar coordinates and the opposite

[pic 1]

[pic 2]Hidalgo State University[pic 3]

City High School

Sahagun

Mechanical Engineering

Vector Calculus

Transformation from rectangular to polar coordinates and the opposite

Reyes Gonzalez Roman Alfieri 449470

Sainos Muños Jorge David 339402

Vázquez Hérnandez Vivian 413409

Vazquez Montaño Alan Yair 383065

Date: 13/05/2022   Group: 2   Qualification:

Teacher: Barrera Gonzales Francisco Javier

Transformation from rectangular to polar coordinates and the opposite

Objective

Obtain information in a precise way and the necessary means to be able to convert two-dimensional data to three-dimensional data to locate movements in a work environment in an artificial plane by means of calculations to real physical movements.

Generate cases in which it is necessary to change the dimensions and therefore the coordinates from rectangular to polar by means of calculations and mathematical formulas.

Conceptual Framework

Introduction

Polar coordinates are written in the form (r, θ), where r is the distance and θ is the angle. These coordinates can be related to rectangular or Cartesian coordinates using trigonometry, a right triangle, and the Pythagorean theorem. It turns out that we use the tangent function to find the angle and the Pythagorean theorem to find the distance, r.

How to transform from rectangular coordinates to polar coordinates?

We recall that rectangular coordinates are written in the form (x, y) and polar coordinates are written in the form (r, \theta), where r is the distance from the origin to the point and θ is the angle formed along the line and the x-axis. These coordinates are related using trigonometry.

Using the right triangle, we can obtain relationships for the polar coordinates in terms of the rectangular coordinates. We note that the x-coordinates form the base of the right triangle and the y-coordinates form the height.

Furthermore, we see that the distance r corresponds to the hypotenuse of the triangle. So we can use the Pythagorean theorem to find the length of the hypotenuse: [pic 4]

The angle θ can be found using the tangent function. Remember that the tangent of an angle is equal to the opposite side divided by the adjacent side. The opposite side is the y component and the adjacent side is the x component. So, we have:   *xy][pic 5]

Since the range of the inverse tangent function is from  to , this does not cover all four quadrants of the Cartesian plane, so many times, the calculator may give the wrong value of .[pic 6][pic 7][pic 8]

Rectangular Coordinates

Modeling

The modeling system that governs rectangular coordinate systems has always been rooted in the "x" and "y" axes since these axes can be put in a cartesian plane and can be divided by means of 4 compartments that can be dividing by 360 degrees and dividing by 4 reduces to 90 degrees following the sexagecinal system regularly, although that does not exclude other degree systems such as radians. Based on the conceived divisions, it can be called as:

• (X, Y) Occurs when the X coordinates are positive and at the same time the Y coordinates are positive.[pic 9]
• (-X, Y) Occurs when the X coordinates are negative and at the same time the Y coordinates are positive.
• (-X, -Y) Occurs when the X coordinates are negative and at the same time the Y coordinates are negative.
• (X, -Y) Occurs when the X coordinates are positive and at the same time the Y coordinates are negative. 

Practical use[pic 10]

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