The rational cohomology of the moduli space T_g of trigonal curves of genus g is known for low genera. It has been computed by Looijenga for g=3 and by Tommasi for g=4. In the first part of this seminar, I will give a full description of the rational cohomology of T_5. First, we will recall the definition and the main properties of trigonal curves of genus g. We will also give some motivation for the study of the cohomology of their moduli spaces. Then we will show how to compute this invariant when g=5. This will be done by studying the embedding of genus 5 trigonal curves in the 1st Hirzebruch surface and by using Gorinov-Vassiliev’s method.