Radian

[Primarygun]

How does radian bring convenience to us? Could anyone tell me?

[psi20]

Radians in trigonometry are used in trignometric graphs, sectors, angular speed, arc lengths, polar graphs, etc.

 

They might also be a mnemonic thing. In the unit circle, the angles with a denominator of 4 have sqrt 2 / 2 (2*2 = 4)as their coordinates and you have to think which quadrant they're in. denominators of 6 have sqrt 3 / 2 as x-coordinate (3*2 = 6) . denominators of 3 have 1/2 as x-coordinate (1+2=3?).

 

That may/may not be a convienience. I'm not good with mnemonics, though, so that doesn't apply to me.

[matt grime]

The convenience is that if the argument is in radians then differentiating sin gives cos, etc. If it were degrees then the derivative of sin would be something like (2pi/360)cos

[Primarygun]

Is it useless?

[matt grime]

Of course it's not useless (no more so than any other unit). It is analytically (ie in terms of calculus) and geometrically the correct measure of angle. It simply and directly relates angle to arc length and sector area.

Degrees are the convenient one - their sole benefit is to divide the circle in to units such that 2,3,4,5,6,8,9,10,12 divide the whole - it serves no other use than this numerical one.

[Sayonara]

Haven't we done this before?

 

[123rock]

Radians aren't units

[123rock]

Radians aren't units

[Primarygun]

Is degree a unit?

[Primarygun]

Is degree a unit?

[swansont]

Is degree a unit?

 

No. I think of degrees and radians as pseudounits - you need to keep track of them to be consistent, but they are an implied ratio (fractions of a circle) and as such don't really have any dimension.

[swansont]

Is degree a unit?

 

No. I think of degrees and radians as pseudounits - you need to keep track of them to be consistent, but they are an implied ratio (fractions of a circle) and as such don't really have any dimension.

[123rock]

Since a degree is measured by the length of the arc divided by the radius of a circle, they aren't unites, since it's distance/distance.

[123rock]

Since a degree is measured by the length of the arc divided by the radius of a circle, they aren't unites, since it's distance/distance.

[NSX]

I think the idea of radians is useful, because it relates the arc length, radius, and the angle subtended. In addition (like Matt said), it's useful in Calculus because degrees don't work there.

 

In addition, steradians use the same idea as radians, whereas relating steradians to degrees would be different.

[NSX]

I think the idea of radians is useful, because it relates the arc length, radius, and the angle subtended. In addition (like Matt said), it's useful in Calculus because degrees don't work there.

 

In addition, steradians use the same idea as radians, whereas relating steradians to degrees would be different.

[CPL.Luke]

why don't degrees work in calculus?

[Dave]

It's not that they won't work, rather that the answer you get is rather ugly.

 

When you differentiate/integrate a trigonometric function, the answer turns out to be nice because we assume the angle is being measured in radians - e.g. sin differentiates to cos. If we were to use degrees when differentiating, we'd have 180s and pi's floating about everywhere.

[CPL.Luke]

yeah thats what I thought, i was starting to think I was doing something seriously wrong in my calc

[123rock]

I think that the reason there is a great deal of use for the radian is because since the unit circle has a radius of 1 (for ratio purposes), the circumference is 2pi and thus 2pi radians for both the degree measure and the length of the circumference. I realize that this is a false derivation of the use of radians, since even if you double the radius of the circle, you double the length of the circumference, but the degree measure is still 360 thus 2pi, since all circles are similar.

 

That's why 180deg.=pi radians, and then they eventually got that 1 radian=about 52degrees.