Trigonometry question

[Steven G]

There is a trig concept that I never really understood and am hoping that someone here can explain it me. For the record other tried and all have failed in explaining this to me. This time I have a group of people who will try to help me in case one person can't do this alone. I really need to resolve this in my mind.

Suppose, for example, we want to calculate an arc length of a circle where the radius is 3 inches and the arc is 1/6 of the circle. So the arc is 60 degree. I completely understand that the arc length can't be (3 inches)* (60 degrees) because what would the meaning of inch*degree mean.

Lets move away from the answer to my problem for a moment. When it comes to degrees, 70 and 60 are not degrees. Rather 70 degrees and 60 degrees are degrees. Similarly 7 and pi are not radians. Rather 7 radians and pi radians are radian measures.

Back to my problem. So 60 degrees = pi/3 radians. So why is the arc length S= 3inches * (pi/3) = pi inches and not pi inch*radian. I have been told that radians have no units but I get confused since I thought the units are radians. What am I missing? Thanks, Steve

 

[Otis]

The radian is a ratio of two lengths, so the units cancel (similar to trig ratios).

?

 

[Harry_the_cat]

OK here goes ...

The circumference of a circle is 2 * pi * r, yes? Do you have the same problem with that formula?

If r was in inches, you are multiplying by 2pi (which effectively is a scalar with no units), so 2*pi*r will be in inches too. You are not multiplying by 2pi radians.

Similarly, if you have an angle of 60 degrees, you have 60/360 ie 1/6 of a circle so the arc length is (1/6)*2*pi*r = (pi/3)*r.

Does that help?

 

[Dr.Peterson]

So your actual question is this:

So why is the arc length S= 3inches * (pi/3) = pi inches and not pi inch*radian. I have been told that radians have no units but I get confused since I thought the units are radians. What am I missing? Thanks, Steve

It's a tricky question; there are multiple ways it could be explained. I recalled discussing it before, and found this from 17 years ago: Are Angles Dimensionless?

There I surveyed other sources to get different perspectives. If my opinion isn't clear, maybe the others will help.

 

[Steven G]

The radian is a ratio of two lengths, so the units cancel (similar to trig ratios).

?

This might be helpful, but I do not see the two lengths which we are finding the ratio of. I think of radians as one way of measuring an angle. I do not see any ratio, sorry.

 

[Steven G]

OK here goes ...

The circumference of a circle is 2 * pi * r, yes? Do you have the same problem with that formula?

If r was in inches, you are multiplying by 2pi (which effectively is a scalar with no units), so 2*pi*r will be in inches too. You are not multiplying by 2pi radians.

Similarly, if you have an angle of 60 degrees, you have 60/360 ie 1/6 of a circle so the arc length is (1/6)*2^pi*r = (pi/3)*r.

Does that help?

The circumference of a circle is 2 * pi * r, yes? Do you have the same problem with that formula? No I do not. I can derive that formula and pi just happens to come up. pi~3.14...., just some real number.

Similarly, if you have an angle of 60 degrees, you have 60/360 ie 1/6 of a circle so the arc length is (1/6)*2*pi*r = (pi/3)*r. Ok, I kinda agree that (1/6)*2*pi*r = (pi/3)*r. I would normally 100% agree except in the situation we are taking about. I agree the lhs has the units of r and so does the rhs. I still disagree that it should be pi/3 on the rhs. I claim (and realize that somehow I am wrong) that it should be (pi/3) radians.

Suppose we are looking at an angle and I say oh that angle looks like (pi/3) radians would I be wrong in what I said? That is, why sometimes you can say radians and other times you can't?

 

[Dr.Peterson]

This might be helpful, but I do not see the two lengths which we are finding the ratio of. I think of radians as one way of measuring an angle. I do not see any ratio, sorry.

The radian measure of an angle is defined as the ratio of arc length to radius. An arc of length 1 m, in a circle with radius 1 m, has measure 1 m/1 m = 1 [radian].

Using the word "radian" amounts to identifying what ratio you are using.

 

[Harry_the_cat]

The circumference of a circle is 2 * pi * r, yes? Do you have the same problem with that formula? No I do not. I can derive that formula and pi just happens to come up. pi~3.14...., just some real number.

Similarly, if you have an angle of 60 degrees, you have 60/360 ie 1/6 of a circle so the arc length is (1/6)*2*pi*r = (pi/3)*r. Ok, I kinda agree that (1/6)*2*pi*r = (pi/3)*r. I would normally 100% agree except in the situation we are taking about. I agree the lhs has the units of r and so does the rhs. I still disagree that it should be pi/3 on the rhs. I claim (and realize that somehow I am wrong) that it should be (pi/3) radians.

Suppose we are looking at an angle and I say oh that angle looks like (pi/3) radians would I be wrong in what I said? That is, why sometimes you can say radians and other times you can't?

If radians don't come into the formula 2*pi*r then how come they suddenly come into the formula (1/6)*2*pi*r ?

 

[Harry_the_cat]

If you are measuring an angle or stating the size of an angle then you need to state the unit you are measuring in, ie 60 degrees or pi/3 radians.

 

[Steven G]

OK, I understand now. The link that Dr Peterson helped me resolve it along with looking at the definition. Thanks!

 

[pka]

The radian measure of an angle is defined as the ratio of arc length to radius. An arc of length 1 m, in a circle with radius 1 m, has measure 1 m/1 m = 1 [radian]. Using the word "radian" amounts to identifying what ratio you are using.

I have a different take on the idea of radian measure: Radian describes the plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc. I follow Ed Moise on this (he is an R L Moore PhD) and Harvard professor.
An angle of one radian measure, if a central angle of any circle then intercepts an are of that circle of length equal to the radius of the circle. Thus the term radian derives from radius. In a unit circle a central angle of measure one radian intercepts an arc of length one.

 

[Steven G]

I never bothered to look at the definition of a radian as I heard my teachers always say 6 radians, pi radians etc so I was sure that radians were the units.
Thanks