# A Simple Introduction to Polar Coordinates | Formulas, Examples & Practice Tasks

Polar coordinates! A word that sounds tricky and difficult but believe me, it’s easy peasy lemon squeezy. Generally, polar coordinates is a system of locating points in a plane with a reference point O.

In this article, we will give you the basic understanding of a Simple Introduction to Polar Coordinates | Formulas, Examples & Practice Tasks. Moreover, you will see the practical implementations of polar coordinates.

**What is a Polar Coordinate? **

**Pole – the reference point**

Polar coordinate is a method to explain the location of a point in a plane. It’s different from the Cartesian plane coordinates which use the x and y axis. Polar coordinates use a distance from a fixed point and an angle from a fixed direction. The coordinates are written in r and (θ).

- Radius (r) – A distance from a fixed point to the origin.

- Angle (θ) – It’s the angle made between a positive x-axis and the line connecting the origin to the point.

In simple words, each and every point on a plane of a two-dimensional coordinate system is evaluated by a distance from a reference point and an angle taken from a reference direction is called polar coordinate system.

**Polar Coordinate Formula **

We can write and find infinite numbers of polar coordinates for a single coordinate point. By using this formula:

(r, ô + 2πn) or (-r, o + (2n + 1) π)

- n is representing the integer.

**Thetha’s**value will be positive if it is measured counterclockwise

**Thetha’s**value will be negative if it is measured clockwise.

- The value of r will be positive if taken from the terminal side of (θ).

- The value of r will be negative if taken from the origin side of theta.

**Important Note **

The side of the plane where the angle contempt is called the initial side.

**Why use polar coordinates? **

Normally, we use polar coordinates specifically in the scenarios where the relationship between points is more naturally expressed in terms of angles and distances.

**For instance;**

They are frequently used in physics, engineering and navigation.

For more information and knowledge read my recent article Reflexive Property in Geometry | Congruence, Proof & Examples

**How to plot points in polar coordinates? **

To plot a point in polar coordinates, you need to understand radius and angle.

**For instance;**

The point (5,45°) is 5 units away from the origin. It forms an angle 45° with the positive x-axis.

**How to convert between polar and Cartesian coordinates? **

It’s a basic understanding. You must know about the conversion between Cartesian and polar coordinates. It’s a step-by-step guide to convert between polar and Cartesian coordinates.

**From polar to Cartesian **

To convert polar coordinates (r,θ) to Cartesian coordinates (x,y):

x = r cos (θ)

y = r sin (θ)

**From Cartesian to polar**

To convert Cartesian coordinates (x,y) to polar coordinates (r, θ):

r = √x^2 + y^2

θ = tan^-1 (x/y)

**Real-World Examples of Polar Coordinates**

**Navigation**

Imagine I am a sailor and trying to navigate in the ocean. Instead of saying “go 5 miles east and 6 miles north” (Cartesian), you can say “sail 5 miles at an angle of 53 degrees” (polar).

**Spiral Pattern**

Polar coordinates are perfect for explaining spirals. For example, the equation r=θr = \theta=θ explains a spiral where the radius increases as the angle increases.

For more information and knowledge read my recent article Exploring the Geometry Of Circles: Tangents, Secants & Chords With Examples

**Fun Facts about Polar Coordinates **

- Like polar bears live near the poles. Polar coordinates revolve around the pole or origin point.

- We use polar coordinates to express art and craft such as: they are used in computer graphics to create patterns.

- Stars and planets are plotted through polar coordinates. People who believe in astronomy used polar coordinates to express stars in the sky.

**Practice tasks of polar coordinates**

To practice problems of polar coordinates, you must try solving these.

**Problem : 1**

Convert the Cartesian coordinates (3,4)(3, 4) to polar coordinates.

r = √3^2 + 4^2

r = √9 + 16

r = √25

r = 5

θ = tan^-1 (4/3)

θ = 53.13°

So, the polar coordinates are 5, 53.13°.

**Problem : 2 **

Convert the polar coordinates (7,60°) to Cartesian coordinates.

x = 7 cos (60°)

7*0.5 = 3.5

y = 7 sin (60°)

7*√3/2 = 6.06

So, the Cartesian coordinates are 3.5, 6.06.

**Tips to learn about polar coordinates **

**Practice, Practice, and Practice**: The more you work with polar coordinates, the more comfortable you’ll become. Practice the above -mentioned problems and change the values.**Use Visual Aids**: Graphing polar coordinates can help you understand the relationships between radius, angle, and position. Radius, angle and origin are the 3 basic understandings one must have.**Remember the Quadrants**: Keep in mind which quadrant you are in to ensure the angle θ is correct.

**Final Words **

It’s a powerful way to express the position of points in a plane. From describing navigation to designing digital prints or exploring the knowledge about stars and planets, it will help you throughout mathematics. Understanding polar coordinates can open up a world of possibilities. Practice is a key to success. Stay curious and learn new inventions about polar coordinates.

**Frequently asked questions **

**What is the effect of polar coordinates?**

It’s a useful tool to create a distorted effect of an image or a video. It can switch the position of the characters along the x-axis and y-axis.

**Can a polar coordinate be a negative? **

Yes, if the measurements of an angle is taken in a clockwise direction, the angle will be negative.

**Is there any benefit of polar coordinates? **

Yes, there are many benefits of polar coordinates in mathematics and the real-world. You can use the polar coordinates formula to solve equations and problems. You can use polar coordinates to describe the navigation and draw the digital image of structures.