Tricky Answers
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From: Ray Vickson on 31 Jul 2010 18:07 On Jul 31, 12:42 am, William Elliot <ma...(a)rdrop.remove.com> wrote: > On Fri, 30 Jul 2010, Ray Vickson wrote: > > On Jul 29, 9:56 pm, William Elliot <ma...(a)rdrop.remove.com> wrote: > >> What are the solutions to the equations > > >> sin^-1 x = 1/sin x, > > >> cos^-1 x = 1/cos x? > > >> Since > >> sin^-2 x = 1/sin^2 x, > > > If sin^-1 x means arcsin(x), then what the heck does sin^-2 x mean? Is > > it arcsin(arcsin(x))? > > Does sin^2 x = sin sin x? No, but that is only because the notation sin^2 is somewhat standard. This would not be the case for arcsin^2: since that notation is not at all standard, I would perfectly well be allowed to define it as arcsin^2 x = arcsin(arcsin x) or as (arcsin x)^2. However, if I were going to use it in a document, I would certainly define it first. Please, just tell me: what is so hard about defining what you mean? R.G. Vickson > > >> x can be anything not in pi.Z. > > >> Since > >> cos^-2 x = 1/cos^2 x? > >> x can be anything not in pi(Z + 1/2). > >
From: Virgil on 31 Jul 2010 19:28 In article <30ca2a57-710d-4f40-9153-26eed41dc0ea(a)n19g2000prf.googlegroups.com>, Ray Vickson <RGVickson(a)shaw.ca> wrote: > On Jul 31, 12:42�am, William Elliot <ma...(a)rdrop.remove.com> wrote: > > On Fri, 30 Jul 2010, Ray Vickson wrote: > > > On Jul 29, 9:56�pm, William Elliot <ma...(a)rdrop.remove.com> wrote: > > >> What are the solutions to the equations > > > > >> sin^-1 x = 1/sin x, > > > > >> cos^-1 x = 1/cos x? > > > > >> Since > > >> sin^-2 x = 1/sin^2 x, > > > > > If sin^-1 x means arcsin(x), then what the heck does sin^-2 x mean? Is > > > it arcsin(arcsin(x))? > > > > Does sin^2 x = sin sin x? > > No, but that is only because the notation sin^2 is somewhat standard. > This would not be the case for arcsin^2: since that notation is not at > all standard, I would perfectly well be allowed to define it as > arcsin^2 x = arcsin(arcsin x) or as (arcsin x)^2. However, if I were > going to use it in a document, I would certainly define it first. > Please, just tell me: what is so hard about defining what you mean? > > R.G. Vickson One way to avoid the confusion is always to write 1/sin(x) as csc(x) or as sin(x)^(-1) and the inverse function to sin(x) on -pi < x <= pi as arcsin(x) thus avoiding 'sin^-1 x' entirely
From: Mike Terry on 31 Jul 2010 19:34 "Ray Vickson" <RGVickson(a)shaw.ca> wrote in message news:f29a1f14-e065-4d02-9c2e-4f2855621da0(a)q21g2000prm.googlegroups.com... > On Jul 31, 12:42 am, William Elliot <ma...(a)rdrop.remove.com> wrote: > > On Fri, 30 Jul 2010, Ray Vickson wrote: > > > On Jul 29, 9:56 pm, William Elliot <ma...(a)rdrop.remove.com> wrote: > > >> What are the solutions to the equations > > > > >> sin^-1 x = 1/sin x, > > > > >> cos^-1 x = 1/cos x? > > > > >> Since > > >> sin^-2 x = 1/sin^2 x, > > > > > If sin^-1 x means arcsin(x), then what the heck does sin^-2 x mean? Is > > > it arcsin(arcsin(x))? > > > > Does sin^2 x = sin sin x? > > No, but that is not the issue: YOU used the notation sin^-2 x, not me. > Why can't you just say what you mean? Is that so hard? Obviously William means that sin^-2 x = (sin x)^-2 and sin^-1 x = (sin x)^-1 etc. I believe this may actually be the "standard" interpretation of the notation he used(?). Looking at his "tricky equations" thread, it seems that he has "tricked" everyone, since everyone interpreted sin^-1 x as arctan x. And he even announced in advance he was trying to trick everyone! Ho ho ho ho ho. Well done William. (Of course, it could be argued that it's not a particularly amusing trick, but it's quite good compared to William's sense of humour in other threads! :-) You could start a long discussion about what the notation sin^-1 x "actually" means and argue with him, but why spoil the joke... Mike. > > R.G. Vickson > > > > > > >> x can be anything not in pi.Z. > > > > >> Since > > >> cos^-2 x = 1/cos^2 x? > > >> x can be anything not in pi(Z + 1/2). > > > > >
From: Virgil on 31 Jul 2010 23:37 In article <h_qdnb6_CKidLsnRnZ2dnUVZ8hCdnZ2d(a)brightview.co.uk>, "Mike Terry" <news.dead.person.stones(a)darjeeling.plus.com> wrote: > "Ray Vickson" <RGVickson(a)shaw.ca> wrote in message > news:f29a1f14-e065-4d02-9c2e-4f2855621da0(a)q21g2000prm.googlegroups.com... > > On Jul 31, 12:42 am, William Elliot <ma...(a)rdrop.remove.com> wrote: > > > On Fri, 30 Jul 2010, Ray Vickson wrote: > > > > On Jul 29, 9:56 pm, William Elliot <ma...(a)rdrop.remove.com> wrote: > > > >> What are the solutions to the equations > > > > > > >> sin^-1 x = 1/sin x, > > > > > > >> cos^-1 x = 1/cos x? > > > > > > >> Since > > > >> sin^-2 x = 1/sin^2 x, > > > > > > > If sin^-1 x means arcsin(x), then what the heck does sin^-2 x mean? Is > > > > it arcsin(arcsin(x))? > > > > > > Does sin^2 x = sin sin x? > > > > No, but that is not the issue: YOU used the notation sin^-2 x, not me. > > Why can't you just say what you mean? Is that so hard? > > Obviously William means that > > sin^-2 x = (sin x)^-2 > and sin^-1 x = (sin x)^-1 etc. > > > I believe this may actually be the "standard" interpretation of the notation > he used(?). Looking at his "tricky equations" thread, it seems that he has > "tricked" everyone, since everyone interpreted sin^-1 x as arctan x. And he > even announced in advance he was trying to trick everyone! Interpreting sin^-1 x as arctan x instead of arcsin x would be indeed tricky!!!
From: jmorriss on 1 Aug 2010 00:15
On Jul 31, 7:28 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <30ca2a57-710d-4f40-9153-26eed41dc...(a)n19g2000prf.googlegroups.com>, > Ray Vickson <RGVick...(a)shaw.ca> wrote: > > > > > > > On Jul 31, 12:42 am, William Elliot <ma...(a)rdrop.remove.com> wrote: > > > On Fri, 30 Jul 2010, Ray Vickson wrote: > > > > On Jul 29, 9:56 pm, William Elliot <ma...(a)rdrop.remove.com> wrote: > > > >> What are the solutions to the equations > > > > >> sin^-1 x = 1/sin x, > > > > >> cos^-1 x = 1/cos x? > > > > >> Since > > > >> sin^-2 x = 1/sin^2 x, > > > > > If sin^-1 x means arcsin(x), then what the heck does sin^-2 x mean? Is > > > > it arcsin(arcsin(x))? > > > > Does sin^2 x = sin sin x? > > > No, but that is only because the notation sin^2 is somewhat standard. > > This would not be the case for arcsin^2: since that notation is not at > > all standard, I would perfectly well be allowed to define it as > > arcsin^2 x = arcsin(arcsin x) or as (arcsin x)^2. However, if I were > > going to use it in a document, I would certainly define it first. > > Please, just tell me: what is so hard about defining what you mean? > > > R.G. Vickson > > One way to avoid the confusion is always to write > 1/sin(x) as csc(x) or as sin(x)^(-1) and > the inverse function to sin(x) on -pi < x <= pi as arcsin(x) > thus avoiding 'sin^-1 x' entirely- Hide quoted text - > > - Show quoted t Uh... So, does sin(x)^(-1) equal csc(x), or does it equal sin(1/x) ? I agree about the arcsin.. My job of teaching math would be MUCH easier if the -1 exponent on a trig function only meant the reciprocal... While we're at it, we should limit the use of inverse, as well... |