why is plane angle measured in radians?

Question

if we consider the eqn, s=r *theta(angle) then
theta = s/r...since the arc and radius is measured in metres...both get cancelled n hence theta should have been dimensionless ie with no unit...

Answer 1

Yes, that is quite correct. Any measure of an angle, strictly speaking, is a ratio of two lengths (for radians, the ratio of the arc to the radius, and for degrees, the ratio of the arc to 1/360th of the circumference) and is thus a dimensionless quantity. The signifier of radians is not a unit for dimensional analysis purposes, but merely an indicator of which two lengths it is the ratio of. It would indeed be correct to refer to an angle of, say, π, rather than π radians, and this is done frequently in mathematics contexts (where angles, like everything else, are just numbers that may be entered into functions), but somewhat less frequently in physics contexts, where people are used enough to seeing units following every measurement that if confronted with "an angle of π" would immediately say "π what?" So we say π radians to avoid that sort of confusion.

Answer 2

no my friend a radian is not the unit of angle
a radian is given as the ratio of 1 m arc to 1 m radius of any circle

we use radians just to differentiate it from DEGREES

Answer 3

A radian was never a unit to begin with. We call it a radian but a radian isn't a unit

Answer 4

i'm no longer extremely helpful what you recommend via accepted trigonometric ratio for -(pi)/2= (0,-a million) i'm doing this gfom memory so...i'm no longer extremely helpful what the accepted trignometric ratio for 7(pi)/4 seem at your unit circle. i'm exceptionally helpful its ((sq. root of)2 /"2",-(sq. root of)2 /"2")