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I got excellent grades in high-school math. I took a year of math in college. I knew how to calculate with radians, but never understood their basis until vie

• derekp7 13y ago
  So here's a question -- is it just the animation that helped, or is it the combination of the animation and being 20 years older? Or, more specifically, having the luxury of being able to throw brain power at the subject without having to worry about learning material for several other subjects, and many sub-topics within all those subjects, all at the same time.
  • reuven 13y ago
    I'm sure that it all helped. But I definitely remember asking, on several occasions, why we would ever want to use radians rather than degrees, and why they seemed to be in such weird increments. And no one ever really told me. The animation was helpful in that it explained everything very quickly and obviously, obviating the need for a numeric or verbal explanation.
• dbbolton 13y ago
  I believe when we were first introduced to the radian, it was explicitly defined in terms of the radius, i.e. 1 radian = an angle whose arc length is equal to the radius of the circle. Was a radian just defined as an angle of 57.3 degrees in your class? I imagine that would seem really arbitrary to students.
• thaumasiotes 13y ago
  > I knew how to calculate with radians, but never understood their basis until viewing this animation, despite asking people on a number of occasions what the benefits were vs. degrees. (They never gave me a real answer to that question, and certainly never derived the basis for radians.) Well, the way I think of the basis for radians goes like so: 1. All circles are similar to all other circles (assumed without proof), so there's a relationship between the radius of a circle and its circumference. 2. Angles, being scale-free, ignore the size of a particular circle; if an angle captures 1/3 of a small circle, it will capture the same 1/3 of a larger one, or a smaller one. 3. We can use the combination of an angle ("How much of the circle are we talking about?") and a radius ("How big is this circle?") to specify an arclength along the edge. We do it by multiplying the radius by the angle, and then by a conversion factor. 4. The radian measure is chosen so that the conversion factor mentioned in step (3) is equal to 1, which is easy to multiply by. Here's the problem: out of a normal-sized group of randomly selected math students, expect between 0 and 1 of them (generally 0) to be interested in hearing you say this. There's another argument for using radians (related, but not obviously so, to the first), which goes like this: 1. When measuring in radians, the trig functions behave well when taking the derivative: d/dx (sin x) = cos x and so forth. 2. If we used degrees (e.g., call dsin x the sine function for degrees instead of radians), we'd get all these awful constants mucking up our computations. d/dx (dsin x) = (180/π) dcos x, and d/dx (dtan x) = (180/π) dsec^2 x. The fourth derivative of sin x is sin x, but the fourth derivative of dsin x is (1049760000/π^4) dsin x. So we use radians to measure angles for the same reason we use e to measure logarithms -- they are more correct. That one has the double disadvantage of being something no one wants to hear and drawing on applications they haven't yet heard of. There's a shorter, less accurate version of the first approach I presented, that goes like so: 1. A circle's radius and circumference are both linear measures; they both have units of length to the first power. 2. Therefore, the conversion between them is a dimensionless (= unitless) number, a constant. 3. Radians are the unique correct constant for that "conversion" (there can't be more than one, any more than there can be more than one constant that converts meters to kilometers). This is short and easy to say, but is sloppier and unsatisfying to precisely the students who care.