Question

Find the principal value of the following:

`tan^-1(-1/sqrt3)`

Answer 1

We have `tan^-1(-1/sqrt3)=-tan^-1(1/sqrt3)`    `[because tan^-1(-x)=-tan^-1x]`

Let `tan^-1(1/sqrt3)=y`

Then,

`tany=1/sqrt3`

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

Thus,

`tany=1/sqrt3=tan(pi/6)`

`=>y = pi/6`

`therefore tan^-1(-1/sqrt3)=-tan^-1(1/sqrt3)`

= - y

`=-pi/6in(-pi/2,pi/2)`

Hence, the principal value of `tan^-1(-1/sqrt3)  is   -pi/6.`