How do you find the exact value of #Arctan (-1) #?

1 Answer
A. S. Adikesavan
Apr 25, 2016

#arc tan(-1)=npi+(3pi)/4, n=0, +-1, +-2, +-3...# n=0, gives its value as #(3pi)/4 in [0, pi]#. Including n=1, we get two values #(3pi)/4 and (7pi)/4in [0, 2pi]#.

Explanation:

Use #tan (pi-x)=tan x#.
#tan(pi-pi/4)=-tan(pi/4)=-1#

So, a principal value of #arc tan (-1) in [0, pi] is (3pi)/4#.
There is another #in [0, 2pi]. tan(2pi-pi/4)=-tan (pi/4)=-1#

So, there are two, #(3pi)/4 and (7pi)/4, in [0, 2pi]#.

Either can be taken as principal value, for the general value

#arc tan(-1)=npi+(3pi)/4, n=0, +-1, +-2, +-3...# . .

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