The meaning of half of a 1/3 cup in cooking is 1/2 * 16 tsp = 8 tsp.
To get half of a 1/3 cup take a measuring cup that as 1/3 marking on it if you have one that does and fill it halfway before it reaches 1/3 cup.
Then you have half of a 1/3 cup.
Or just use 8 tablespoons to measure 1/2 of a 1/3 cup.
Having measuring cups and measuring spoons come in really handy to get the exact measurements when it comes to cooking and recipes.
There are also 16 tablespoons in 1 cup of ingredients whether dry ingredients or liquid ingredients.