How do you evaluate #tan^-1(-1)# without a calculator?

1 Answer
Dec 1, 2016

Depending upon if the domain is restricted to #((-pi)/2,pi/2)# the solution will be #pi/4# or if the domain is not restricted the solution will be #pi/4+pik# where #k# is an interger #>=1#

Explanation:

What you are trying to solve is

#x=tan^-1(-1)#

Since the problem deals with the inverse tangent, that means we will use the tangent function to solve the problem.

#tan(x)=tan(tan^-1(1))#

#tan(x)=1#

Now let's think about when #tan(x)=1#

If we are rotating an angle and dealing with a right triangle, we know that #tan(45^@)=1/1=1# and #tan(225^@)=(-1)/-1=1#

Since you are more than likely discussing radians, #x=pi/4#, #x=(5pi)/4#, and we can continue to add #pik# where #k# is an integer#>=1#

Therefore #x=pi/4+pik# where #k# is an interger #>=1#.

If the domain is restricted to #((-pi)/2,pi/2)# then the solution will be #x=pi/4#.