Convert Cartesian Coordinates to Polar Coordinates
When calculating polar coordinates, the conversion from Cartesian coordinates involves the use of trigonometric functions. The distance (r) is determined by the square root of the sum of the squared x and y coordinates, while the angle (θ) is found using the tangent function.
Converting Cartesian coordinates (x, y) to polar coordinates involves determining the radial distance (r) and the angle (θ). The conversion is performed using the following formulas:
Radial distance: The radial distance is the straight-line distance from the origin (0,0) to the point (x, y).
r = √(x2 + y2)
Angle (θ): The angle θ is the tangent of the ratio of the y-coordinate to the x-coordinate. Note that special attention should be given to the quadrant in which the point lies to obtain the correct angle.
θ = tan(y/x)
Example: Covert the point P(3, 4) from Cartesian coordinate system to polar coordinates.
Solution:
Step 1: Identify the coordinates of the point in the Cartesian system.
- x = 3
- y = 4
Step 2: Use the following formulas to convert Cartesian to Polar Coordinates:
r = √(x2 + y2), and
θ = tan(y/x)
Step 3: Substitute the values of x and y into the formulas:
r = √(x2 + y2) = √(32 + 42) = √(9 + 16) = √25 = 5, and
θ = tan(4/3)
Step 4: Calculate the angle in degrees using a calculator:
≈ 53.13°
Step 5: Write the Polar Coordinates:
(P(3, 4) in Cartesian Coordinates is equivalent to P(5, 53.13°) in Polar Coordinates.
Therefore, the Polar Coordinates of the point P(3, 4) are P(5, 53.13°).
Plotting Points in Polar Coordinates
Graph is plotted for (r, θ) for plotting Points in Polar Coordinates
For polar coordinates (r = 5, θ = 45°)
Polar co-ordinates are plotted as below:
Polar Coordinates System
The polar coordinate system is a two-dimensional coordinate system that employs distance and angle to represent points on a plane. It’s similar to a regular coordinate system, but instead of using x and y coordinates, it uses:
- Radius (r): The distance from a fixed reference point, known as the origin or pole.
- Angular coordinate (θ): The angle measured counterclockwise from a fixed direction, referred to as the polar axis.
Key features of the polar coordinate system:
- Points are identified with an ordered pair (r, θ). An example would be the point (2, π/3), meaning it lies 2 units away from the origin while maintaining an angle of π/3 (or approximately 60 degrees) from the polar axis.
- The angle θ ranges between 0 and 2π (360 degrees). However, negative angles are valid too; these simply imply moving counterclockwise past the polar axis.
- Variable ‘r’, representing the radius, accepts non-negative values only. When r equals 0, it indicates that the point sits right on top of the origin.
In this article, we will discuss the Polar Coordinate System in detail, including its Properties, Graph, Formula, and Examples.
Table of Content
- What is Polar Coordinate System?
- What are Polar Coordinates?
- Graph of Polar Coordinates
- Polar Coordinates Formula
- Cartesian to Polar Coordinates Conversion
- Polar to Cartesian Coordinates Conversion
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