The angles m π/n (with m, n integers) for which the trigonometric functions may be expressed in terms of finite root extraction of real numbers are limited to values of m which are precisely those which produce constructible polygons. Analytic expressions for trigonometric functions with arguments of this form can be obtained using the Wolfram Language function ToRadicals, e.g., ToRadicals[Sin[Pi/17]], for values of n>=7 (for n<=6, the trigonometric functions auto-evaluate in the Wolfram Language).
257-gon | 65537-gon | constructible polygon | Fermat prime | heptadecagon | Morrie's law | pentagon | Sierpiński sieve | trigonometry angles--0 | trigonometry angles--pi | trigonometry angles--pi/10 | trigonometry angles--pi/11 | trigonometry angles--pi/12 | trigonometry angles--pi/13 | trigonometry angles--pi/15 | trigonometry angles--pi/16 | trigonometry angles--pi/17 | trigonometry angles--pi/18 | trigonometry angles--pi/2 | trigonometry angles--pi/20 | trigonometry angles--pi/23 | trigonometry angles--pi/24 | trigonometry angles--pi/3 | trigonometry angles--pi/30 | trigonometry angles--pi/32 | trigonometry angles--pi/4 | trigonometry angles--pi/5 | trigonometry angles--pi/6 | trigonometry angles--pi/7 | trigonometry angles--pi/8 | trigonometry angles--pi/9 | unit circle
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