Solved Questions on Polar Coordinates
Question 1: Convert Cartesian to Polar Coordinates: Given (x, y) = (3, 4), find the polar coordinates (r, θ).
Solution:
Given: (x, y) = (3, 4)
Thus, x = 3 and y = 4
Using formula r = √{x2 + y2} and θ = tan-1(y/x), we get
r = √{32 + 42} = 5, and
θ = tan-1(4/3)
Question 2: Convert Cartesian to Polar Coordinates: Given (x, y) = (6, 8), find the polar coordinates (r, θ).
Solution:
Given: (x, y) = (6, 8)
Thus, x = 6 and y = 8
Using formula r = √{x2 + y2} and θ = tan-1(y/x), we get
r = √{62 + 82} = 10, and
θ = tan-1(8/6)
Question 3: Convert Polar to Cartesian Coordinates: Given (r, θ) = (2, π /4), find the Cartesian coordinates (x,y).
Solution:
Given: (r, θ) = (2, π /4)
Thus, x = 2 and y = π /4
Using formula x = rcos(θ) and y = rsin(θ)
x = 2cos( π /4)
y = 2sin( π /4)
Question 4: Convert Polar to Cartesian Coordinates: Given (r, θ) = (6, π/3), find the Cartesian coordinates (x,y).
Solution:
Given: (r, θ) = (6, π /3)
Thus, x = 6 and y = π /3
Using formula x = rcos(θ) and y = rsin(θ)
x = 6cos( π /3)
y = 6sin( π /3)
Question 5: Convert Cartesian to Polar Coordinates: Given (x, y) = (8, 15), find the polar coordinates (r, θ).
Solution:
Given: (x, y) = (8, 15)
Thus, x = 8 and y = 15
Using formula r = √{r2 + y2} and θ = tan-1(y/x), we get
r = √{82 + 152} = 17, and
θ = tan-1(15/8)
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