Question

Find the principal value of the following:

`tan^-1(2cos  (2pi)/3)`

Answer 1

Let `tan^-1(2cos  (2pi)/3)=y`

Then,

`tany=2cos  (2pi)/3`

We know that the range of the principal value branch is `(-pi/2,pi/2)`

Thus,

`tany=2cos  (2pi)/3 =2xx(-1)/2=-1=tan(-pi/4)`

`=>y=-pi/4in(-pi/2,pi/2)`

Hence, the principal value of `tan^-1(2cos  (2pi)/3)    is    -pi/4.`