algebraic form of any trygonometrical function

**hellbike** · Nov16-09, 06:21 AM

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

**Hurkyl** · Nov16-09, 06:36 AM

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

$LaTeX Code: \\sin \\pi y = 1/10$

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** · Nov16-09, 06:48 AM

sin 17 = x

I want to know algebraic form of x

(its 17 grades)

**Hurkyl** · Nov16-09, 08:30 AM

Originally Posted by hellbike

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** · Nov16-09, 09:44 AM

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

$LaTeX Code: 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 $

**Hurkyl** · Nov16-09, 09:50 AM

I assume he meant 17 gradians, which is $LaTeX Code: 17 \\pi / 200$ 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)