Preheat oven to 375°.
Lightly spoon flour into a dry measuring cup; level with a knife. Combine flour, baking soda, and cream of tartar, stirring with a whisk.
Combine 1 cup of sugar and butter in a large bowl, and beat with a mixer at medium speed until well blended. Add the corn syrup, vanilla, and egg; beat well. Gradually add the flour mixture to the sugar mixture, beating just until combined. Cover and chill for 10 minutes.
Combine 3 tablespoons sugar and cinnamon, stirring with a whisk.
With moist hands, shape dough into 42 (1-inch) balls. Roll balls in sugar mixture. Place balls 2 inches apart onto baking sheets coated with cooking spray. Flatten balls with the bottom of a glass. Bake at 375° for 5 minutes (cookies will be slightly soft). Cool on baking sheets for 2 minutes. Remove cookies from pans; cool completely on wire racks.