Hi Piet. Thank you for the good comment. I agree with your observation if the interior angle were just theta, i.e. a measurable angle, that wasn't small. It can be shown that even if theta were as "large" as 30 degrees, the percentage difference between theta and sin(theta) is less than 5%. That said, however, in this video, I am not dealing even with a "measurable" angle "theta", but rather, a "differential" angle "d"-"theta" which is infinitesimally small to avoid any such conflicts :))
I like this proof. It is a companion proof to dividing the circle into an infinitely large number of pie slices, taking half of those slices and reassembling the pieces into a rectangle which in the limit as theta appraches zero the dimensions of the rectangle are r and pi.
I realized that you get into circular reasoning ;-) when you already know the area of a wedge is equalt to half the angle times r squared, with a 2 pi angle you already have the result you want. Mostly people will use the triangle first and then state the infinitissimal becomes equal to the wedge angle.
I had a physics teacher that always said whenever using infinitesimals (usually for integrals of a function): "Is the function constant in this area? If not, make the area smaller". This helped me understand a lot of things with infinitesimals and it gave me the intuition behind it. (there isn't really a function here, but the idea is that with infinitesimals you can imagine dTheta to be really small such that the arc given by dTheta is just a straight line)
I think you should mention that you can only calculate the area of the right triangle and not the wedge. Area of the triangle is 1/2 sin(theta)r *r. For small area's, especially infenitissmals sin(theta) equals theta, so you can substitute.
You have to also teach the students that the height of the slice 'approaches' r and that arclength approaches the base of the triangle ..as dedetha approaches zero.
Hi Piet. Thank you for the good comment. I agree with your observation if the interior angle were just theta, i.e. a measurable angle, that wasn't small. It can be shown that even if theta were as "large" as 30 degrees, the percentage difference between theta and sin(theta) is less than 5%. That said, however, in this video, I am not dealing even with a "measurable" angle "theta", but rather, a "differential" angle "d"-"theta" which is infinitesimally small to avoid any such conflicts :))
I like this proof. It is a companion proof to dividing the circle into an infinitely large number of pie slices, taking half of those slices and reassembling the pieces into a rectangle which in the limit as theta appraches zero the dimensions of the rectangle are r and pi.
I realized that you get into circular reasoning ;-) when you already know the area of a wedge is equalt to half the angle times r squared, with a 2 pi angle you already have the result you want. Mostly people will use the triangle first and then state the infinitissimal becomes equal to the wedge angle.
I had a physics teacher that always said whenever using infinitesimals (usually for integrals of a function): "Is the function constant in this area? If not, make the area smaller". This helped me understand a lot of things with infinitesimals and it gave me the intuition behind it. (there isn't really a function here, but the idea is that with infinitesimals you can imagine dTheta to be really small such that the arc given by dTheta is just a straight line)
I think you should mention that you can only calculate the area of the right triangle and not the wedge. Area of the triangle is 1/2 sin(theta)r *r. For small area's, especially infenitissmals sin(theta) equals theta, so you can substitute.
thank you! it was very helpful when you described dtheta as the "interior angle", it made sense after that!
0riginal Kati Glad to hear it was helpful.
This was so helpful. Thank you, sir.
You have to also teach the students that the height of the slice 'approaches' r and that arclength approaches the base of the triangle ..as dedetha approaches zero.
Good one -- I like your humor about "circular reasoning" .. sort of like a "chicken/egg" argument, so I appreciate your insight on this. Thank you ;)
very nice ... thank you very much.
Truly appreciate your kind note .. glad the video was a help!
thank you
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