Sift flour with 3/4 cup of sugar. Set aside.
Beat egg whites, cream of tartar, salt and vanilla until stiff enough to form soft peaks. I lift the beaters and look at the little mounds that form. For soft peaks, they should stand up with just the tops of the peaks folding over, kind of like the top of a soft-serve ice cream cone.
Add the remaining 3/4 cup of sugar a little at a time while continuing to beat until stiff peaks form. For stiff peaks, the peaks should be standing up completely.
Sift about 1/4 cup of the flour mixture over the egg whites and gently fold in. Repeat about four more times until all the flour mixture is added, folding in each addition gently.
Bake in a ungreased 10 inch tube pan (I don't have a tube pan, so I used a bundt pan) and bake at 375 degrees for 35 to 40 minutes.
Invert cake in pan to cool completely before removing the cake. Some angel food tube pans have feet that allow them to rest inverted (cake side down) without setting it directly on a plate. This prevents steam from forming and making the cake soggy. If your pan doesn't have the feet, invert it on a bottle with the neck of the bottle fitting in the center hole. You can also use four soup or similar cans or cups spaced so that the rim of the pan rests on the cans/cups.
Once it has cooled completely, remove from the pan by gently running a thin spatula or knife around the edge of the pan and around the tube, being careful not to cut the cake. Invert over a plate and it should come out. If you used a bundt pan like I did, it may take a few tries and possibly gently pulling the cake away from the sides of the pan.