odd functions

• Feb 4th 2007, 07:17 AM
gracy
odd functions
Show that arctan x is an odd function, that is, arctan –x = –arctan x.
• Feb 4th 2007, 09:11 AM
CaptainBlack
Quote:

Originally Posted by gracy
Show that arctan x is an odd function, that is, arctan –x = –arctan x.

Let \$\displaystyle x=\tan(y)\$, now \$\displaystyle \tan\$ is an odd function so:

\$\displaystyle -x=\tan(-y)\$,

so:

\$\displaystyle
\arctan(-x)=-y
\$

but \$\displaystyle y=\arctan(x)\$, hence:

\$\displaystyle
\arctan(-x)=-\arctan(x)
\$

so \$\displaystyle \arctan\$ is an odd function.

RonL
• Feb 4th 2007, 09:34 AM
ThePerfectHacker
Quote:

Originally Posted by gracy
Show that arctan x is an odd function, that is, arctan –x = –arctan x.

In general if \$\displaystyle f\$ is invertible and odd.
Then \$\displaystyle f^{-1}\$ is invertible and odd.
• Feb 4th 2007, 11:41 AM
topsquark
I'm not objecting to CaptainBlack's proof as much as I have a question about how to get around a problem with it. Given an angle x define y:
\$\displaystyle y = tan(x)\$
Then
\$\displaystyle atn(y) = atn(tan(x)) \neq x\$ in general because of the domain restriction we place on the atn function to make it bijective.

I can easily see how restricting the domain of the tan function would fix this, but then we aren't really using the tan function. The only way I can think of to get around THIS is to extend the atn function so that it's no longer 1:1. But then it isn't really the inverse of the tan function any longer.

I'm kinda going in circles here...

Just wondering if it wouldn't be better to prove that atn is an odd function by using something more direct.

-Dan
• Feb 4th 2007, 01:03 PM
CaptainBlack
Quote:

Originally Posted by topsquark
I'm not objecting to CaptainBlack's proof as much as I have a question about how to get around a problem with it. Given an angle x define y:
\$\displaystyle y = tan(x)\$
Then
\$\displaystyle atn(y) = atn(tan(x)) \neq x\$ in general because of the domain restriction we place on the atn function to make it bijective.

I can easily see how restricting the domain of the tan function would fix this, but then we aren't really using the tan function. The only way I can think of to get around THIS is to extend the atn function so that it's no longer 1:1. But then it isn't really the inverse of the tan function any longer.

I'm kinda going in circles here...

Just wondering if it wouldn't be better to prove that atn is an odd function by using something more direct.

-Dan

There is an implicit restriction so that arctan takes value in the interval (pi/2,pi/2), and the domain of tan is also restricted to the same interval.

(especialy as arctan is not odd on any other open interval of length pi).

RonL