Tricky Answers
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From: Virgil on 1 Aug 2010 01:24 In article <37526874-54ab-47bb-89d6-84ca71e6086f(a)e20g2000vbn.googlegroups.com>, "jmorriss(a)idirect.com" <jmorriss(a)idirect.com> wrote: > 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) ? While sin(x^-1) = sin(1/x) works, sin(x)^(-1) = sin(1/x) does not! > > 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...
From: William Elliot on 1 Aug 2010 01:58 On Sat, 31 Jul 2010, Virgil wrote: > 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? > > 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 > That's what my mathematical handbook uses to avoid this notational ambiguity. Here's another. Solve sin^2 x = sin^2 x, cos^2 x = cos^2 x. Huh? That's the problem? (sin x)^2 = sin^2 x = sin sin x (cos x)^2 = cos^2 x = cos cos x ----
From: William Elliot on 1 Aug 2010 02:11
On Sun, 1 Aug 2010, Mike Terry wrote: > "Ray Vickson" <RGVickson(a)shaw.ca> wrote in message >>> On Fri, 30 Jul 2010, Ray Vickson 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! > Shucks, they missed my pun. Here's another. Solve sin^2 x = sin^2 x and cos^2 x = cos^2 x. > 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... There's no arguing with a pun. |