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Cosines of Common Angles—C.E. Mungan, Fall 2008 There is a pattern to the cosines of the commonly memorized angles, as nicely diagrammed in the unit circle on the middle of the Wikipedia page about the unit circle: φ cos φ 30˚ 3 2 45˚ 2 2 60˚ 1 2 We can improve on the trend by adding in two more angles: φ cos φ 0˚ 4 2 30˚ 3 2 45˚ 2 2 60˚ 1 2 90˚ 0 2 This looks nice, except the angles don’t divide up the first quadrant evenly. Let’s add in 15˚ and 75˚, whose cosines are easily worked out using the half-angle formulae: φ cos φ 0˚ 2+ 4 2 15˚ 2+ 3 2 30˚ 2+ 1 2 45˚ 2+ 0 2 60˚ 2− 1 2 75˚ 2− 3 2 90˚ 2− 4 2 Great, the first quadrant is uniformly divided into 15˚ segments. But the symmetry of the cosine values is now imperfect: What happened to 2+ 2 2 and 2− 2 ? 2 As you can easily check, they’re equal to cosine of 22.5˚ and 67.5˚! It’s no coincidence that the second table in this note (which had a uniform pattern to the cosine values) was symmetric about 45˚, and now we’re finding a symmetry about half of that angle (i.e., 22.5˚) when we again write down a uniform pattern of cosine values. To see the trend take the inverse cosine of the following angle in degrees and don’t be too surprised at what you get: 2+ 2+ 2 . 2