You Do Need power plug adapter while traveling from Cook Islands to Dominica.

Here is Why You Need An Adapter

The outlet and plug type used in Dominica are D and G and in Cook Islands is I. This is how they look like.

Cook Islands

In Cook Islands, outlets of type I are in use.

Type I

Dominica

In Dominica, outlets of type D and G are in use.

Type D

Type G