I was sitting the other day with my cousin, helping him with his homework, when he asked me what's the deal with these radians.
So I took a paper and a pen and tried to recall where did these radians come from, but something is not quite clear for me.
Consider the following circle:
When the angle is 90 degrees, arc a is exactly the length of the full circle's perimeter.
and therfor the angle in radians is:
So when r=1, we're good, we get:
But obviously, it is not the case for any other r, as it should be...
So what am I missing here?
Thanks in advanced!
Hi Plato, thanks for your post.
The property you mentioned (that the change in radius doesn't affect the angle) is understood (it's quite intuitive actually), but still, we need to somehow define radian.
Do we arbitrarily define the radius to be 1, when we talk about radians?
Equivalently, the radian measure of an angle is the length of the arc of circle the angle subtends divided by the radius of the circle. Of course, if the radius is 1, that reduces to "the length of the arc". But if the circle has radius "R" then it has circumference . One fourth or the circle has circumference so that the radian measure of this 90 degree angle is .