Question

What is t he exact value of csc(arctan-1/3)) ?

what is t he exact value of csc(arctan-1/3)) ?

Answer 1

`csc(arctan(-1/3))=-sqrt10`

Explanation:

Let `arctan(-1/3)=A`, then `tanA=-1/3` and hence `A` is in interval `-pi/2< A < 0` i.e. `A` is in `Q4`

and as we are seeking `cscA` and it is negative in `Q4`

As `tanA=-1/3`, `cotA=1/(-1/3)=-3`

and `cscA=-sqrt(1+cot^2A)=-sqrt(1+(-3)^2)=-sqrt10`

hence `csc(arctan(-1/3)=-sqrt10`