Preheat the oven to 350 degrees.
Combine the butter and ¾ cup of the sugar in the bowl of an electric mixer. With the paddle attachment, cream the butter on medium speed until light and fluffy. Add the egg and beat until smooth. Stir together the flour, cream of tartar, baking soda, and salt. Fold the flour mixture into the butter mixture.
Mix together the remaining 1½ tablespoons sugar and the cinnamon. Form the dough into 1-inch balls and roll them in the sugar mixture. Line a baking sheet with parchment paper and put the cookies several inches apart on the pan.
Bake the cookies for 8 to 10 minutes, until golden brown. Allow them to cool for 5 to 10 minutes and then remove them from the baking sheet.