Question

Evaluate the following:

`tan^-1 1+cos^-1 (-1/2)+sin^-1(-1/2)`

Answer 1

Let `sin^-1(-1/2)=y`
Then,
`siny=-1/2`
We know that the range of the principal value branch is `[-pi/2,pi/2].`
Thus,

`siny=-1/2=sin(-pi/6)`

`=>y=-pi/6in[-pi/2,pi/2]`

Now,
Let cos^-1(-1/2)= z
Then,
`cosz=-1/2`
We know that the range of the principal value branch is [0, π].
Thus,

`cosz=-1/2=cos((2pi)/3)`

`=>z = (2pi)/3in[0,pi]`
so

`tan^-1 1+cos^-1(-1/2)+sin^-1(1/2)=pi/4+(2pi)/3-pi/6=(3pi)/4`

`therefore tan^-1 1+cos^-1(-1/2)+sin^-1(1/2)=(3pi)/4`