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## Main Question or Discussion Point

sin x = 1/10, or any other number that can't be found in math tables - how to know what x is?

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- #1

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sin x = 1/10, or any other number that can't be found in math tables - how to know what x is?

- #2

Hurkyl

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sin x = 1/10

is sufficient to "know" x.If by "know" you mean a decimal approximation, well it's easy enough to get that from a calculator with the arcsine function.

If by "know" you mean finding an algebraic number

[tex]\sin \pi y = 1/10[/tex]

then I very, very strongly suspect it's impossible -- y would have to be transcendental, not algebraic.*: An algebraic number is any number that is the root of a polynomial with integer coefficients. Numbers that can be expressed in terms of integers, +, -, *, /, and taking of roots are kinds of algebraic numbers.

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sin 17 = x

I want to know algebraic form of x

(its 17 grades)

I want to know algebraic form of x

(its 17 grades)

- #4

Hurkyl

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You almost surely don't want to know anything aboutsin 17 = x

I want to know algebraic form of x

(its 17 grades)

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According to Maple, sine 17 degrees is a zero of the polynomial

[tex]

281474976710656\,{x}^{48}-3377699720527872\,{x}^{46}+18999560927969280

\,{x}^{44}

-66568831992070144\,{x}^{42}+162828875980603392\,{x}^{40}-295364007592722432\,{x}^{38}

+411985976135516160\,{x}^{36}-

452180272956309504\,{x}^{34}+396366279591591936\,{x}^{32}

-280058255978266624\,{x}^{30}+160303703377575936\,{x}^{28}-74448984852135936\,{x}^{26}

+28011510450094080\,{x}^{24}-8500299631165440\,{x}^{22}+2064791072931840\,{x}^{20}

-397107008634880

\,{x}^{18}+59570604933120\,{x}^{16}-6832518856704\,{x}^{14}+583456329728\,{x}^{12}

-35782471680\,{x}^{10}+1497954816\,{x}^{8}-39625728\,{x}^{6}

+579456\,{x}^{4}-3456\,{x}^{2}+1

[/tex]

- #6

Hurkyl

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I'm pretty sure this requires solving a quintic too. (Only one quintic -- sin 72 can be expressed in terms of square roots)

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