- #1
hellbike
- 61
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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?
Algebraic form of any trygonometrical function
In summary, the equation sin x = 1/10 can be used to find x for many purposes, but it is impossible to find an algebraic form of x that satisfies the equation sin x = 1/10, as x would have to be transcendental. Additionally, the equation sin 17 = x is only useful for finding the approximate value of x, and finding the algebraic form of x for this equation would require solving a quintic polynomial.
- #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
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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
\,{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]
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 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)
1. What is the algebraic form of a trigonometric function?
The algebraic form of a trigonometric function is a mathematical expression that represents the relationship between the sides and angles of a right triangle. It uses variables and constants to represent the different parts of the triangle, such as the hypotenuse, opposite, and adjacent sides.
2. How is the algebraic form of a trigonometric function different from the graphical form?
The algebraic form of a trigonometric function is a mathematical representation of the relationship between the sides and angles of a right triangle, while the graphical form shows this relationship visually on a coordinate plane. The algebraic form is more precise and can be used to calculate exact values, while the graphical form provides a visual understanding of the function.
3. Why is it important to know the algebraic form of a trigonometric function?
Knowing the algebraic form of a trigonometric function allows for the calculation of exact values and can be used to solve equations involving trigonometric functions. It also provides a deeper understanding of the relationship between the sides and angles of a right triangle.
4. Can all trigonometric functions be written in algebraic form?
Yes, all trigonometric functions can be written in algebraic form. However, some functions may be more complex and require the use of multiple trigonometric identities to be expressed algebraically.
5. How can I convert a trigonometric function from graphical form to algebraic form?
To convert a trigonometric function from graphical form to algebraic form, you can use the Pythagorean identity and other trigonometric identities to rewrite the function in terms of the variables and constants representing the sides and angles of a right triangle. You can also use a graphing calculator or other software to help visualize and express the function algebraically.
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