Implied domain

**Stroodle** · March 13th 2010, 04:19 AM

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

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

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

But apparently this is wrong...

Thanks for your help.

**Grandad** · March 13th 2010, 04:46 AM

Hello Stroodle

Originally Posted by Stroodle

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

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

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

But apparently this is wrong...

Thanks for your help.

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

Grandad

**Stroodle** · March 13th 2010, 04:55 AM

Ahh, now I get it. Thanks for your help.

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

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

Thanks again.

**Grandad** · March 13th 2010, 10:53 AM

Hello Stroodle

Originally Posted by Stroodle

Ahh, now I get it. Thanks for your help.

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

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

**Stroodle** · March 13th 2010, 01:13 PM

Oh. Our teacher taught us that $Tan(x)$ is the same as $tan(x)$ except that its domain is restricted to $x\in \left (-\frac{\pi}{2},\frac{\pi}{2}\right )$

**Krizalid** · March 13th 2010, 01:46 PM

what a ridiculous thing, it's just a matter of upper and lowercase, it's just absurd.

**Grandad** · March 13th 2010, 11:01 PM

Has anyone else come across this 'convention'?

Grandad

**Stroodle** · March 13th 2010, 11:06 PM

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 $tan(x)$ was restricted to $x\in \left ( -\frac{\pi}{2},\frac{\pi}{2}\right )$ would my original answer be correct?

By the way, I had a look at your paintings Grandad. They are amazing

**Grandad** · March 14th 2010, 12:53 AM

Hello Stroodle

Originally Posted by Stroodle

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 $tan(x)$ was restricted to $x\in \left ( -\frac{\pi}{2},\frac{\pi}{2}\right )$ 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

$\text{Tan }(2\arcsin(x))$

would indeed be $\left ( -\frac{\pi}{2},\frac{\pi}{2}\right )$

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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