Exercise $4.16$ from Fulton-Harris' book on representation theory is stated as follows: If $g$ is a $d$-cycle in the symmetric group $S_d$, then show that $\chi_{\lambda}(g) = (-1)^s$ if $\lambda$ is a hook of shape $(d-s, 1, \dots, 1)$, and $0$ if $\lambda$ is not a hook.
I have solved the second part via orthogonality relations. How can I show that $\chi_{\lambda}(g) = (-1)^s$ if $\lambda$ is a hook of the desired shape? I know I must use Frobenius' character formula, but I am unsure how to manipulate it.