Question

​Find the principal values of the following:

`cos^-1(sin   (4pi)/3)`

Answer 1

Let `cos^-1(sin   (4pi)/3) = y`

Then,

`cosy = sin  (4pi)/3`

We know that the range of the principal value branch is [0, π].

Thus, 

`cosy=sin  (4pi)/3=-sqrt3/2=cos    (5pi)/6`

`=>y=(5pi)/6in[0,pi]`

Hence, the principal value of `cos^-1(sin  (4pi)/3)`  is  `(5pi)/6`.