inverse tan[tan(6pi/7)] i need an exact answer using pi as needed. can someone explain to me how to solve this?

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inverse tan[tan(6pi/7)] i need an exact answer using pi as needed. can someone explain to me how to solve this?

Trigonometry
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Sorry I mean \[\huge \arctan\left( \tan (6\pi/7) \right) \] or \[\huge \tan^{-1}\left( \tan (6\pi/7) \right) \]
\[\tan^{-1} [\tan (6\pi/7)]\]
What do you know about inverse functions? (they're super important to answering this question).

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um i know how to put them in my calculator and that they have restrictions this particular one the answer must be in either quadrant 1 or 4...
The key is that an inverse function "undoes" a function. arctan ( tan ( stuff ) ) = stuff and tan ( arctan ( stuff ) ) = stuff (assuming domains and ranges are valid) also \(\large \arctan(x) = \tan^{-1}(x) \) it's just notation. They mean the same exact thing.
so would the answer just be 6pi/7 since the inverse tan would undo tan?
  • phi
yes
thanks (: does that apply to any function that has the inverse of the same function on the outside?
6pi/7 or -pi/7, they are the same thing. And yes, inverse function ( function ( stuff ) ) = stuff function ( inverse function ( stuff ) ) = stuff Inverse functions and functions "undo" each other. That's glossing over a lot of details, but that's the big idea. :)
  • phi
yes, and vice versa.
what if it was something like \[\tan [\tan^{-1}(\pi/10)] \]
the simplified version of 6pi/7 would be just pi/7 ?
  • phi
yes, pi/10 you can check with your calculator tan^-1 (pi/10) you are treating pi/10 as a number not as an angle.
  • phi
the simplified version of 6pi/7 would be just pi/7 ? I would not say "simplified" If you mean Is pi/7 the "reference angle" of 6pi/7, that is true
the problem says simplify and use pi as needed..
  • phi
In that case, the answer is 6 pi/7 (about 154.29ยบ)
  • phi
unless they want the answer between -pi/2 and pi/2 in which case they want -pi/7 (see math teacher's post)
ok thank you sooo much ! i have test on this tomorrow and i had no clue how to do it! you are a lifesaver!

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