Preheat the oven to 375°F. Line two baking sheets with parchment paper, if you have it, otherwise leave it ungreased. In a medium bowl, stir together the flour, cream of tartar, baking soda, and salt.
In a large bowl, using an electric mixer, beat together the butter and ¾ cup of the sugar for about 2 minutes, until light and fluffy, scraping down the sides as needed. Beat in the egg until completely incorporated, then beat in the vanilla extract. With the mixer on low, beat in the flour mixture until incorporated, scraping down the sides as needed.
Place the remaining ¼ cup sugar and the cinnamon in a shallow bowl and stir to combine. Troll the batter into 1 ½-inch balls, then roll the balls in the sugar mixture, working in batches. Place them on the baking sheets, at least 2 inches apart. If you like a flatter cookies, press down on the balls with your hand or the bottom of a glass.
Bake them for 10 to 12 minutes, rotating the sheets midway if you are baking more than one sheet at a time, so that both sheets have a chance to be on the upper rack.
Transfer the snickerdoodles to a wire rack to cool.