- #1
hellbike
- 61
- 0
sin x = 1/10, or any other number that can't be found in math tables - how to know what x is?
Algebraic form of any trygonometrical function
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Discussion Overview
The discussion revolves around finding the algebraic form of trigonometric functions, specifically focusing on the sine function for angles that are not standard values found in mathematical tables, such as 1/10 and 17 gradians. Participants explore the implications of expressing these values algebraically and the complexities involved in doing so.
Discussion Character
- Exploratory
- Technical explanation
- Mathematical reasoning
- Debate/contested
Main Points Raised
- One participant questions how to determine the value of x when sin x = 1/10, prompting a discussion on what "know" means in this context.
- Another participant suggests that for many purposes, the equation sin x = 1/10 is sufficient, but obtaining a decimal approximation can be done using a calculator's arcsine function.
- A participant expresses skepticism about finding an algebraic number y such that sin πy = 1/10, suggesting it would likely be transcendental rather than algebraic.
- Multiple participants mention that sin 17 gradians (or degrees) is not straightforward to express algebraically, with one noting it requires solving a cubic equation and others referencing polynomial equations related to sine values.
- One participant provides a specific polynomial that sine 17 degrees is a zero of, indicating the complexity of the algebraic representation.
- Another participant clarifies that 17 gradians is equivalent to 17π/200 radians, suggesting that this angle may also require solving a quintic equation for its sine value.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the algebraic representation of sine values for non-standard angles. There are competing views on the complexity involved in expressing these values algebraically, with some suggesting polynomial equations and others indicating the potential transcendental nature of certain sine values.
Contextual Notes
The discussion highlights the limitations of expressing trigonometric functions algebraically, particularly for angles that are not multiples of standard angles, and the dependence on polynomial equations that may not have straightforward solutions.
- #2
Hurkyl
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What do you mean by "know"? For a great many purposes, the equation
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
*: 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.
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
[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.
- #3
hellbike
- 61
- 0
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 about x aside from the fact it is the sine of 17 gradians, and is approximately 0.26387305. What are you trying to do?hellbike said:
- #5
g_edgar
- 606
- 0
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
[tex] 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[/tex]
According to Maple, sine 17 degrees is a zero of the polynomial
[tex] 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[/tex]
- #6
Hurkyl
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I assume he meant 17 gradians, which is [itex]17 \pi / 200[/itex] 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)
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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