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Gauss' trigonometric identities for heptadecagon


Today, we write down Gauss' magical trigonometric identities for regular heptadecagon.


$$\cos{\frac{\pi}{17}} = \frac{1}{16} \left( 1 - \sqrt{17} + \sqrt{34 - 2 \sqrt{17}} + 2 \sqrt{17 + 3 \sqrt{17} + \sqrt{34 - 2 \sqrt{17}} + 2 \sqrt{34 + 2 \sqrt{17}}} \right)$$
$$\cos{\frac{2 \pi}{17}} = \frac{1}{16} \left( -1 + \sqrt{17} + \sqrt{34 - 2 \sqrt{17}} + 2 \sqrt{17 + 3 \sqrt{17} - \sqrt{34 - 2 \sqrt{17}} - 2 \sqrt{34 + 2 \sqrt{17}}} \right)$$
$$\cos{\frac{3 \pi}{17}} = \frac{1}{16} \left( 1 + \sqrt{17} + \sqrt{34 + 2 \sqrt{17}} + 2 \sqrt{17 - 3 \sqrt{17} + \sqrt{34 + 2 \sqrt{17}} - 2 \sqrt{34 - 2 \sqrt{17}}} \right)$$
$$\cos{\frac{4 \pi}{17}} = \frac{1}{16} \left( -1 + \sqrt{17} - \sqrt{34 - 2 \sqrt{17}} + 2 \sqrt{17 + 3 \sqrt{17} + \sqrt{34 - 2 \sqrt{17}} + 2 \sqrt{34 + 2 \sqrt{17}}} \right)$$
$$\cos{\frac{5 \pi}{17}} = \frac{1}{16} \left( 1 + \sqrt{17} + \sqrt{34 + 2 \sqrt{17}} - 2 \sqrt{17 - 3 \sqrt{17} + \sqrt{34 + 2 \sqrt{17}} - 2 \sqrt{34 - 2 \sqrt{17}}} \right)$$
$$\cos{\frac{6 \pi}{17}} = \frac{1}{16} \left( -1 - \sqrt{17} + \sqrt{34 + 2 \sqrt{17}} + 2 \sqrt{17 - 3 \sqrt{17} - \sqrt{34 + 2 \sqrt{17}} + 2 \sqrt{34 - 2 \sqrt{17}}} \right)$$
$$\cos{\frac{7 \pi}{17}} = \frac{1}{16} \left( 1 + \sqrt{17} - \sqrt{34 + 2 \sqrt{17}} + 2 \sqrt{17 - 3 \sqrt{17} - \sqrt{34 + 2 \sqrt{17}} + 2 \sqrt{34 - 2 \sqrt{17}}} \right)$$
$$\cos{\frac{8 \pi}{17}} = \frac{1}{16} \left( -1 + \sqrt{17} + \sqrt{34 - 2 \sqrt{17}} - 2 \sqrt{17 + 3 \sqrt{17} - \sqrt{34 - 2 \sqrt{17}} - 2 \sqrt{34 + 2 \sqrt{17}}} \right)$$
$$\cos{\frac{9 \pi}{17}} = \frac{1}{16} \left( 1 - \sqrt{17} - \sqrt{34 - 2 \sqrt{17}} + 2 \sqrt{17 + 3 \sqrt{17} - \sqrt{34 - 2 \sqrt{17}} - 2 \sqrt{34 + 2 \sqrt{17}}} \right)$$
$$\cos{\frac{10 \pi}{17}} = \frac{1}{16} \left( -1 - \sqrt{17} + \sqrt{34 + 2 \sqrt{17}} - 2 \sqrt{17 - 3 \sqrt{17} - \sqrt{34 + 2 \sqrt{17}} + 2 \sqrt{34 - 2 \sqrt{17}}} \right)$$
$$\cos{\frac{11 \pi}{17}} = \frac{1}{16} \left( 1 + \sqrt{17} - \sqrt{34 + 2 \sqrt{17}} - 2 \sqrt{17 - 3 \sqrt{17} - \sqrt{34 + 2 \sqrt{17}} + 2 \sqrt{34 - 2 \sqrt{17}}} \right)$$
$$\cos{\frac{12 \pi}{17}} = \frac{1}{16} \left( -1 - \sqrt{17} - \sqrt{34 + 2 \sqrt{17}} + 2 \sqrt{17 - 3 \sqrt{17} + \sqrt{34 + 2 \sqrt{17}} - 2 \sqrt{34 - 2 \sqrt{17}}} \right)$$
$$\cos{\frac{13 \pi}{17}} = \frac{1}{16} \left( 1 - \sqrt{17} + \sqrt{34 - 2 \sqrt{17}} - 2 \sqrt{17 + 3 \sqrt{17} + \sqrt{34 - 2 \sqrt{17}} + 2 \sqrt{34 + 2 \sqrt{17}}} \right)$$
$$\cos{\frac{14 \pi}{17}} = \frac{1}{16} \left( -1 - \sqrt{17} - \sqrt{34 + 2 \sqrt{17}} - 2 \sqrt{17 - 3 \sqrt{17} + \sqrt{34 + 2 \sqrt{17}} - 2 \sqrt{34 - 2 \sqrt{17}}} \right)$$
$$\cos{\frac{15 \pi}{17}} = \frac{1}{16} \left( 1 - \sqrt{17} - \sqrt{34 - 2 \sqrt{17}} - 2 \sqrt{17 + 3 \sqrt{17} - \sqrt{34 - 2 \sqrt{17}} - 2 \sqrt{34 + 2 \sqrt{17}}} \right)$$
$$\cos{\frac{16 \pi}{17}} = \frac{1}{16} \left( -1 + \sqrt{17} - \sqrt{34 - 2 \sqrt{17}} - 2 \sqrt{17 + 3 \sqrt{17} + \sqrt{34 - 2 \sqrt{17}} + 2 \sqrt{34 + 2 \sqrt{17}}} \right)$$

If you are curious and want to know Gauss' method to derive these formulas, please follow our next few posts.



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