Comments on: When does the sum of three numbers equal their product? http://www.johndcook.com/blog/2008/11/30/tangent-identity/ The blog of John D. Cook Sat, 11 Feb 2012 01:10:06 -0500 http://wordpress.org/?v=2.8.4 hourly 1 By: chandan http://www.johndcook.com/blog/2008/11/30/tangent-identity/comment-page-1/#comment-89627 chandan Sun, 26 Jun 2011 07:59:38 +0000 http://www.johndcook.com/blog/?p=998#comment-89627 i have started loving math after reading these articles..After reading this one, i am thinking whether we can give a geometrical proof for this one? for eg. half of a circle with unit radius is PI radians..just curious thats all.. i have started loving math after reading these articles..After reading this one, i am thinking whether we can give a geometrical proof for this one? for eg. half of a circle with unit radius is PI radians..just curious thats all..

]]> By: Puneet http://www.johndcook.com/blog/2008/11/30/tangent-identity/comment-page-1/#comment-89318 Puneet Fri, 24 Jun 2011 09:45:13 +0000 http://www.johndcook.com/blog/?p=998#comment-89318 Another example : 1+2+3 = 1*2*3 tan inverse 1 = Pi/4 = 45 degrees tan inverse 2 = 63.43 degrees tan inverse 3 = 71.57 degrees ----------------------------------- 180 degrees Another example :
1+2+3 = 1*2*3

tan inverse 1 = Pi/4 = 45 degrees
tan inverse 2 = 63.43 degrees
tan inverse 3 = 71.57 degrees
———————————–
180 degrees

]]> By: Dave Marain http://www.johndcook.com/blog/2008/11/30/tangent-identity/comment-page-1/#comment-14760 Dave Marain Sat, 21 Mar 2009 18:38:45 +0000 http://www.johndcook.com/blog/?p=998#comment-14760 Jonathan, I think John is saying that, even in the trivial case, one could choose (that is, there always exists) angles whose sum is pi. Thus, we can use a = pi, b = pi, c = -pi. Then a+b+c = pi. In other words, we don't have to use angles 0,0 and 0. Jonathan,
I think John is saying that, even in the trivial case, one could choose (that is, there always exists) angles whose sum is pi. Thus, we can use a = pi, b = pi, c = -pi. Then a+b+c = pi. In other words, we don’t have to use angles 0,0 and 0.

]]> By: Jonathan http://www.johndcook.com/blog/2008/11/30/tangent-identity/comment-page-1/#comment-14757 Jonathan Sat, 21 Mar 2009 17:25:25 +0000 http://www.johndcook.com/blog/?p=998#comment-14757 We miss the trivial 0,0,0 solution. We miss the trivial 0,0,0 solution.

]]> By: Math Teachers at Play #3 « JD2718 http://www.johndcook.com/blog/2008/11/30/tangent-identity/comment-page-1/#comment-14756 Math Teachers at Play #3 « JD2718 Sat, 21 Mar 2009 17:24:08 +0000 http://www.johndcook.com/blog/?p=998#comment-14756 [...] follow the link to the sum of three numbers equals their product - that one’s bothering [...] [...] follow the link to the sum of three numbers equals their product – that one’s bothering [...]

]]> By: Dave Marain http://www.johndcook.com/blog/2008/11/30/tangent-identity/comment-page-1/#comment-14713 Dave Marain Thu, 19 Mar 2009 23:46:57 +0000 http://www.johndcook.com/blog/?p=998#comment-14713 John, I just came across this excellent post from the new Carnival: Math Teachers at Play #3. Interestingly, I had recently published the simpler version for two numbers some time ago and even that 'easy' question generated great interest. For your problem, my first reaction was to consider three equal numbers which of course leads to the trivial solution a = b = c = 0 as well as a = b = c = +-sqrt(3). The sqrt(3) might lead one to consider tan(pi/3) which of course is a special case of your general result. I would hope motivated high school students would make that connection and be interested in seeing hte general case! John,
I just came across this excellent post from the new Carnival: Math Teachers at Play #3. Interestingly, I had recently published the simpler version for two numbers some time ago and even that ‘easy’ question generated great interest. For your problem, my first reaction was to consider three equal numbers which of course leads to the trivial solution a = b = c = 0 as well as a = b = c = +-sqrt(3). The sqrt(3) might lead one to consider tan(pi/3) which of course is a special case of your general result. I would hope motivated high school students would make that connection and be interested in seeing hte general case!

]]> By: John Armstrong http://www.johndcook.com/blog/2008/11/30/tangent-identity/comment-page-1/#comment-14710 John Armstrong Thu, 19 Mar 2009 22:40:51 +0000 http://www.johndcook.com/blog/?p=998#comment-14710 I remember this one. I first saw it when my graduate advisor and I were trying to work out a manifestly symmetric proof of Heron's formula. It turns out that this identity is actually equivalent to Heron's formula. I'll leave it as an exercise to supply the two proofs. I remember this one. I first saw it when my graduate advisor and I were trying to work out a manifestly symmetric proof of Heron’s formula. It turns out that this identity is actually equivalent to Heron’s formula. I’ll leave it as an exercise to supply the two proofs.

]]> By: nilo de roock http://www.johndcook.com/blog/2008/11/30/tangent-identity/comment-page-1/#comment-10460 nilo de roock Tue, 02 Dec 2008 21:42:34 +0000 http://www.johndcook.com/blog/?p=998#comment-10460 ( What fascinates me about math is that it always has a next amazement in store. ) You can find the identity in the beautiful book 'Trig or Treat' an Encycopledia of Trigonometric Identity Proofs by Y.E.O. Adrian. ( What fascinates me about math is that it always has a next amazement in store. ) You can find the identity in the beautiful book ‘Trig or Treat’ an Encycopledia of Trigonometric Identity Proofs by Y.E.O. Adrian.

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