How do you find the exact value of arctan(tanx)?

1 Answer
Aug 30, 2015

Answer:

#arctan(tan x) = x - pi floor((x + pi/2)/pi)#

which simplifies to

#arctan(tan x) = x# if #x in (-pi/2, pi/2)#

Explanation:

If #x in (-pi/2, pi/2)# then #arctan(tan x) = x#

Otherwise we need to add some integer multiple of #pi# to #x# to bring it into this range.

Using the floor function, we can write:

#arctan(tan x) = x - pi floor((x + pi/2)/pi)#

Here's a graph of #arctan(tan(x))# :

graph{3pi/5(abs(sin(x/2+pi/4))-abs(cos(x/2+pi/4))-1/6(abs(sin(x/2+pi/4)^3))+1/6(abs(cos(x/2+pi/4)^3)))(tan(x/2+pi/4)/abs(tan(x/2+pi/4))) [-5, 5, -2.5, 2.5]}

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