Preheat oven to 325 degrees F, with rack in the lower third of oven.
Sift together flour and 3/4 cup sugar.
Beat egg whites and warm water with a mixer on low speed until foamy, about 3 minutes. Add cream of tartar, vanilla, salt, and coconut extract.
Increase speed to medium, and beat until soft peaks form, about 3 minutes.
With machine running, add remaining 3/4 cup sugar, a little at a time. Increase speed to high, and beat until peaks are stiff and glossy, about 2 minutes.
Remove bowl from mixer.
Working in 3 batches, sift flour mixture over egg-white mixture, gently folding in each addition with a rubber spatula, being careful not to deflate egg whites.
Slowly pour batter into a 10-inch angel food cake pan.
Bake until cake is golden brown and springs back when touched, 40 to 45 minutes.
Immediately from the oven, carefully invert pan onto a wire rack. Let cake cool in pan, about 1 hour.
Carefully run a knife around sides of cake to loosen, and turn out onto a large plate or cake stand.