Algebraic form of any trygonometrical function

1. Nov 16, 2009

hellbike

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

2. Nov 16, 2009

Hurkyl

Staff Emeritus
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.

3. Nov 16, 2009

hellbike

sin 17 = x

I want to know algebraic form of x

4. Nov 16, 2009

Hurkyl

Staff Emeritus
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?

5. Nov 16, 2009

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

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

6. Nov 16, 2009

Hurkyl

Staff Emeritus
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)