Question

In definition of quantities we know that a quantity has a unit when it has a dimension. According to this definition, while angle does not have a dimension, it should not have a unit, but the unit of angle is radian. How is it possible?

Answer 1

Answer:

Angles measured in radians are considered to be dimensionless because the radian measure of angles is defined as the ratio of two lengths θ=sr (where s is some arc measuring s-units in length, and r is the radius) however the degree measure is not defined in this way and it is said to be dimensionless too.

As long as you choose other units and modify the trigonometric function accordingly, you can actually introduce 'dimensional' quantity measuring the size of an angle. This is quite artificial, however, in the sense that the trigonometric functions in terms of radian have Taylor expansion, e.g.

cosx=∑n=0∞(−1)n(2n)!x2n.

This formula would have contained bunch of unit-cancelling factors if you have chosen a dimensional unit (because we cannot add quantities of different diemsnions). So it is much natural to consider degrees dimensionless.

Explanation:- Imagine you are a surveyor and you have to map a building. You need meters and goniometers for that. Of course, you are not going to use the same tools to measure the perimeter of the parking lot and the size of the tiles in the kitchen floor. For the second one you'll need a smaller meter, and also a smaller unit of measure of length. You need to record that the side of the parking lot measures 10 meters, while the side of a tile measures 10 centimeters: the number 10 alone means nothing, when associated to a length.

On the other hand, even if you are probably going to use a smaller goniometer to measure the angle of the kitchen tile, there will be no need to take note of that. Indeed, the goniometer just tells you how much of a circumference is covered by the angle you are measuring. A right angle covers one quarter of a circumference, and the number 1/4 alone is a perfectly good measure of an angle. That's what "dimensionless" means. "Radians", "degrees", "centi-degrees" are just different ways of writing fractions.