Find the value of: `tan^(-1)(1)+cos^(-1)(-1/2)+sin^(-1)(-1/2)`

Explanation

We know that the ranges of principal values of `tan^(-1),cos^(-1)` and `sin^(-1)` are `(-pi/2,pi/2),[0,pi]` and `[-pi/2,pi/2]` respectively.
Let `tan^(-1)(1)=theta_(2)`. Then,
`costheta_(2) =-1/2 =-cospi/3=cos(pi-pi/3)=cos(2pi)/3`.
`therefore theta_(2)=(2pi)/3 in [0,pi]`.
Let `sin^(-1)(-1/2)=theta_(3)`. Then,
`sintheta_(3)=-1/2=-sinpi/6=sin(-pi/6) rArr theta_(3)=-pi/6 in [-pi/2,pi/2]`.
`therefore tan^(-1)(1)+cos^(-1)(-1/2) + sin^(-1)(-1/2) = (pi/4+(2pi)/3-pi/6) = (3pi)/4`.