Question

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

Answer 1

Sorry I mean

\[\huge \arctan\left( \tan (6\pi/7) \right) \]
or
\[\huge \tan^{-1}\left( \tan (6\pi/7) \right) \]

Answer 2

\[\tan^{-1} [\tan (6\pi/7)]\]

Answer 3

What do you know about inverse functions? (they're super important to answering this question).

Answer 4

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...

Answer 5

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.

Answer 6

so would the answer just be 6pi/7 since the inverse tan would undo tan?

Answer 7

yes

Answer 8

thanks (:
does that apply to any function that has the inverse of the same function on the outside?

Answer 9

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. :)

Answer 10

yes, and vice versa.

Answer 11

what if it was something like \[\tan [\tan^{-1}(\pi/10)] \]

Answer 12

the simplified version of 6pi/7 would be just pi/7 ?

Answer 13

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.

Answer 14

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

Answer 15

the problem says simplify and use pi as needed..

Answer 16

In that case, the answer is 6 pi/7 (about 154.29º)

Answer 17

unless they want the answer between -pi/2 and pi/2
in which case they want -pi/7 (see math teacher's post)

Answer 18

ok thank you sooo much ! i have test on this tomorrow and i had no clue how to do it! you are a lifesaver!