Algebraic form of any trygonometrical function

[hellbike]

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

 

[Hurkyl]

What do you mean by "know"? For a great many purposes, the equation

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^* y such that

$$\sin \pi y = 1/10$$​

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.

 

[hellbike]

sin 17 = x

I want to know algebraic form of x

(its 17 grades)

 

[Hurkyl]

sin 17 = x

I want to know algebraic form of x

(its 17 grades)

You almost surely don't want to know anything about x aside from the fact it is the sine of 17 gradians, and is approximately 0.26387305. What are you trying to do?

 

[g_edgar]

sin of 3 degrees can be written with just square roots, but sine of 17 degrees, 17 not a multiple of 3, requires solving a cubic equation, in addition to several quadratic equations.

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

$$281474976710656\,{x}^{48}-3377699720527872\,{x}^{46}+18999560927969280<br /> \,{x}^{44}<br /> -66568831992070144\,{x}^{42}+162828875980603392\,{x}^{40}-295364007592722432\,{x}^{38}<br /> +411985976135516160\,{x}^{36}-<br /> 452180272956309504\,{x}^{34}+396366279591591936\,{x}^{32}<br /> -280058255978266624\,{x}^{30}+160303703377575936\,{x}^{28}-74448984852135936\,{x}^{26}<br /> +28011510450094080\,{x}^{24}-8500299631165440\,{x}^{22}+2064791072931840\,{x}^{20}<br /> -397107008634880<br /> \,{x}^{18}+59570604933120\,{x}^{16}-6832518856704\,{x}^{14}+583456329728\,{x}^{12}<br /> -35782471680\,{x}^{10}+1497954816\,{x}^{8}-39625728\,{x}^{6}<br /> +579456\,{x}^{4}-3456\,{x}^{2}+1$$

 

[Hurkyl]

I assume he meant 17 gradians, which is $17 \pi / 200$ radians or 15.3 degrees.

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