What+is+a+Radian?

What is a Radian?

The Radian CLIPS storyboarding team has had many discussions around the ways one might think about radians. The currciulum document which says that 'students will recognize the radian as an alternative unit to the degree for angle measurement, define the radian measure of an angle as the length of the arc that subtends this angle at the centre of a unit circle and develop and apply the relationship between radian and degree measure." Many textbooks lead students to the conclusion that 180 degrees = p radians. In order to steer away from this, the writers decided to adopt a ratio definition: Radian measure is the ratio of the arc length, which subtends the central angle, to the radius length of the circle.

Radian Measure = arc length: radius length

We liked the idea of students creating measures based on the radius as the standard unit, examining ratios and families of similar arcs, implicitly seeing that the central angle of such families would be the same, but explicitly referring to the angle.

The development went further to examine families of identical radian measure to be equivalent to the arc length of the unit circle. This was adopted as the standard unit for subsequent student activities. However, the avoidance of referring to the central angle has subsequently led to difficulties as to just what a radian is. Is it a number? or a length? a point on the circle? it is a ratio, hence dimensionless...so what is it measuring?

By avoiding reference to the central angle have we caused ourselves a new set of problems and avoided some truths?

Should we aim at a definition (and if so what?) but also go for different ways of thinking about radians (i.e. length of arc on the unit circle radians, point on the unit circle i.e. /2?); "a radian", "one radian", "radian measure", "radian representation", "common ways of referring to radians and radian measure", radian measure important for application.

Please add your comments to the document started by Myrna last week, which will be synthesized and shared with Peter Taylor, Walter Whitley, Marian & Don Small. Further comments can be posted on the Discussion page.

Thanks, Irene Mar.22, 2010

Robert Mar 20; Donna Mar 22; Dan. Mar. 23; Irene Mar.23



Don Small on March 24 Walter Whiteley March 30 - some reflections on the other measure of angle - the 'turns'.

Peter Taylor, March 31:

Also from **Don Small on March 24** in response to the issue of whether degrees and/or radians are different with respect to having units

If x is in degrees then let X = (2 Pi / 360) x with X in radians. These measure the same angle but in different units. Now you have to write sine differently if you mean the argument is in radians, like sin x = Sin X. The upper case means set the calculator to radians, the lower case, here, will mean use degrees. The equation y = x sin x can be equivalently written Y = (360 / 2 Pi) X Sin X. Lower and upper case y have the same values here. Pick an angle.

Let x be 45 degrees. y = 45 sin(45 degrees) = 45 / sqrt(2) Y = (360 / 2 Pi) ( Pi/4 ) Sin( Pi/4 ) = 45 / sqrt(2)

Both X and x are unit free, if you wish to think that way. They are not really but angle units cannot mix with length units. Engineers use degrees when it is convenient. If they are comparing the phases of AC voltage and they want to connect the Ontario grid to the New York grid, they get the phases synchronized to within a small fraction of a degree before they throw the switch. They don't want another blackout!

I would like to change my mind, if I may. I had indicated that degrees and radians can each be used in any situation involving angles, including calculus. The exception would be when extending the concept of angle to three dimensions, not that this is high school material. I had indicated that radians would be needed in order to form the analogy. Walter Whiteley's contribution of March 30 makes me realize that even here, any units of angle may be used. He suggests a complete rotation, a turn, as a fundamental unit. One may define the measure of a solid angle about a point in space as the area of that portion of the unit sphere that subtends the solid angle. I was using conventional units of area but one could measure the area as the fraction of the sphere. To get an analogy of degrees you could take 1/360 of a sphere, or any other measure you like, 1/(4 Pi) being the one that is presently in use. I believe that radians, degrees and turns are all equally valid units of angle and to assert that one “should” use only one of these, is to perform a disservice.
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