Question

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

Answer 1

`tan^-1(tanpi)=0`.

Explanation:

First of all, recall that `tanpi=0`, so, the reqd. value is `tan^-1(0)`

To find this, we must know the following Defn. of `tan^-1` fun. :

`tan^-1x=theta, x in RR iff tantheta=x, theta in (-pi/2,pi/2)`.

Now, knowing that, `tan0=0, and, 0 in (-pi/2,pi/2)`, we can conclude

from the Defn. that, `tan^-1 0=0, i.e., tan^-1(tanpi)=0`.

It will be interesting to note that, `tan^-1(tanpi)!=pi`.