Why doesn't inverse t a n of t a n ( 124 ) = − 56 degrees instead of 124 degrees? Doesn't the inverse function cancel the tan function?

Question

Why doesn&#039;t inverse   t  a  n of   t  a  n  (  124  )  =  −  56 degrees instead of   124 degrees? Doesn&#039;t the inverse function cancel the tan function?

popman14ee · Accepted Answer

If you look at a graph of   tan  ⁡  (  x  ), you&#039;ll see lots of &quot;up and down&quot; lines, which means the function is not actually invertible - it fails the horizontal line test. To get an inverse, we have to choose one of those lines and flip it across   y  =  x. Mathematicians have agreed to choose the part of   tan  ⁡  (  x  ) which goes through the origin (the part for which   −      π    2    &amp;lt;  x  &amp;lt;      π    2  , or in degrees,   −  90  &amp;lt;  x  &amp;lt;  90). Therefore, the graph of       tan          −      1        ⁡  (  x  ) only takes values between   −      π    2   and       π    2   radians, or   −  90 and   90 degrees. Because of this,       tan          −      1        ⁡  (  tan  ⁡  (  x  )  ) is not always equal to   x.This means you need to do some thinking when you find an angle by taking       tan          −      1      . You have to know whether you are looking for an angle in the first, second, third, or fourth quadrant. The       tan          −      1       function will only give you answers in the first and fourth quadrant, but since   tan  ⁡  (  x  +  π  )  =  tan  ⁡  (  x  ), you can recover answers in the second/third quadrant by adding or subtracting   π to/from the answer you get. (If you insist on working in degrees, then use   180 instead of   π.)

Why doesn't inverse t a n

Answered question

Answer & Explanation

New Questions in Differential Calculus