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Thread: Implied domain

  1. #1
    Senior Member Stroodle's Avatar
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    Implied domain

    Hi there, I'm having a little trouble with working out the implied domain of this equation:

    $\displaystyle y=tan[2Sin^{-1}(x)]$

    I thought that since the range of $\displaystyle 2Sin^{-1}(x)$ must be a subset or equal to the domain of $\displaystyle tan(x)$, then $\displaystyle -\frac{\pi}{2}<2Sin^{-1}(x)<\frac{\pi}{2}$. Making the implied domain $\displaystyle x\in \left (-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right )$

    But apparently this is wrong...

    Thanks for your help.
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  2. #2
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    Grandad's Avatar
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    Hello Stroodle
    Quote Originally Posted by Stroodle View Post
    Hi there, I'm having a little trouble with working out the implied domain of this equation:

    $\displaystyle y=tan[2Sin^{-1}(x)]$

    I thought that since the range of $\displaystyle 2Sin^{-1}(x)$ must be a subset or equal to the domain of $\displaystyle tan(x)$, then $\displaystyle -\frac{\pi}{2}<2Sin^{-1}(x)<\frac{\pi}{2}$. Making the implied domain $\displaystyle x\in \left (-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right )$

    But apparently this is wrong...

    Thanks for your help.
    The range of $\displaystyle \arcsin (x)$ is $\displaystyle \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. So $\displaystyle -\pi \le 2\arcsin(x) \le \pi$. These are therefore potential values for the domain, but you must remember that $\displaystyle \tan \theta$ has problems when $\displaystyle \theta = \pm\frac{\pi}{2}$.

    Grandad
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  3. #3
    Senior Member Stroodle's Avatar
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    Ahh, now I get it. Thanks for your help.

    I actually thought that it was $\displaystyle Tan(x)$ not $\displaystyle tan(x)$.

    If this was the case, would my answer/method have been correct?

    Thanks again.
    Last edited by Stroodle; Mar 13th 2010 at 05:14 AM. Reason: Sentence structure...
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  4. #4
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    Hello Stroodle
    Quote Originally Posted by Stroodle View Post
    Ahh, now I get it. Thanks for your help.

    I actually thought that it was $\displaystyle Tan(x)$ not $\displaystyle tan(x)$.

    If this was the case, would my answer/method have been correct?

    Thanks again.
    Sorry, is this some meaning of Tan that I don't know about? What's the difference between Tan x and tan x ?

    Grandad
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  5. #5
    Senior Member Stroodle's Avatar
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    Oh. Our teacher taught us that $\displaystyle Tan(x)$ is the same as $\displaystyle tan(x)$ except that its domain is restricted to $\displaystyle x\in \left (-\frac{\pi}{2},\frac{\pi}{2}\right )$
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  6. #6
    Math Engineering Student
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    what a ridiculous thing, it's just a matter of upper and lowercase, it's just absurd.
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  7. #7
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    Domain of tan x

    Has anyone else come across this 'convention'?

    Grandad
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  8. #8
    Senior Member Stroodle's Avatar
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    Just did a quick google and found this Complex Analysis/Elementary Functions/Inverse Trig Functions - Wikibooks, collection of open-content textbooks and http://docs.google.com/viewer?a=v&q=...jqEaNUEA&pli=1

    I guess this is what my teacher was talking about...

    Back to my question though; if the domain of $\displaystyle tan(x)$ was restricted to $\displaystyle x\in \left ( -\frac{\pi}{2},\frac{\pi}{2}\right )$ would my original answer be correct?

    By the way, I had a look at your paintings Grandad. They are amazing
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  9. #9
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    Hello Stroodle
    Quote Originally Posted by Stroodle View Post
    Just did a quick google and found this Complex Analysis/Elementary Functions/Inverse Trig Functions - Wikibooks, collection of open-content textbooks and Powered by Google Docs

    I guess this is what my teacher was talking about...

    Back to my question though; if the domain of $\displaystyle tan(x)$ was restricted to $\displaystyle x\in \left ( -\frac{\pi}{2},\frac{\pi}{2}\right )$ would my original answer be correct?

    By the way, I had a look at your paintings Grandad. They are amazing
    You are quite correct - there does seem to be a convention that distinguishes between tan and Tan (and, indeed, sin and Sin, etc).

    The use of the upper-case initial letter, then, restricts the domain of the functions to their principal values. This has the effect of making the functions one-to-one, and therefore makes it possible to define the inverse functions. In that case, the domain of
    $\displaystyle \text{Tan }(2\arcsin(x))$
    would indeed be $\displaystyle \left ( -\frac{\pi}{2},\frac{\pi}{2}\right )$

    Since this upper- and lower-case convention appears not to be very widespread, I should use it with caution if I were you.

    (Thanks for the comment about the paintings. If you just looked at the ones on my profile, you'll find some more in the Chat Room at http://www.mathhelpforum.com/math-he...-painting.html.)

    Grandad
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