Exact trigonometric values
In mathematics, the values of the trigonometric functions can be expressed approximately, as in , or exactly, as in . While trigonometric tables contain many approximate values, the exact values for certain angles can be expressed by a combination of arithmetic operations and square roots. The trigonometric functions of angles that are multiples of 15°, 18°, or 22.5° have simple algebraic values. These values are listed in the following table for angles from 0° to 90°. For angles outside of this range, trigonometric values can be found by applying the reflection and shift identities. In the table below, stands for the ratio 1:0. These values can also be considered to be undefined (see division by zero). {| class="wikitable" style="text-align: center;"
| !Radians!!Degrees!!sin!!cos!!tan!!cot!!sec!!csc |
|---|
| !!! |
| - |
| ! !! |
| - |
| ! !! |
| - |
| ! |
| - |
| ! !! |
| - |
| ! !! |
| - |
| ! !! |
| - |
| ! !! |
| - |
| ! !! |
| - |
| ! |
| - |
| ! !! |
| - |
| ! !! |
| - |
| ! !! |
| } |
| Some exact trigonometric values, such as , can be expressed in terms of a combination of arithmetic operations and square roots. Such numbers are called constructible, because one length can be constructed by compass and straightedge from another if and only if the ratio between the two lengths is such a number. However, some trigonometric values, such as , have been proven to not be constructible. |
| The sine and cosine of an angle are constructible if and only if the angle is constructible. If an angle is a rational multiple of pi radians, whether or not it is constructible can be determined as follows. Let the angle be radians, where a and b are relatively prime integers. Then it is constructible if and only if the prime factorization of the denominator, b, consists of any number of Fermat primes, each with an exponent of 1, times any power of two. |
À propos de ce résultat
Azimuthal decorrelation of jets widely separated in rapidity in pp collisions at $ \sqrt{s}=7 $ TeV
Jian Wang, Matthias Finger, Lesya Shchutska, Qian Wang, Matthias Wolf, Varun Sharma, Konstantin Androsov, Jan Steggemann, Leonardo Cristella, Roberto Castello, Alessandro Degano, Zhirui Xu, Chao Wang, João Miguel das Neves Duarte, Tian Cheng, Yixing Chen, Werner Lustermann, Andromachi Tsirou, Alexis Kalogeropoulos, Andrea Rizzi, Ioannis Papadopoulos, Paolo Ronchese, Thomas Muller, Ho Ling Li, Giuseppe Codispoti, Hua Zhang, Siyuan Wang, Peter Hansen, Daniel Gonzalez, Paul Turner, Wei Sun, Raffaele Tito D'Agnolo, Ji Hyun Kim, Donghyun Kim, Dipanwita Dutta, Zheng Wang, Sanjeev Kumar, Wei Li, Yong Yang, Ajay Kumar, Ashish Sharma, Georgios Anagnostou, Joao Varela, Csaba Hajdu, Muhammad Ahmad, Ekaterina Kuznetsova, Ioannis Evangelou, Matthias Weber, Muhammad Shoaib, Milos Dordevic, Vineet Kumar, Vladimir Petrov, Francesco Fiori, Quentin Python, Meng Xiao, Hao Liu, Sourav Sen, Viktor Khristenko, Marco Trovato, Gurpreet Singh, Fan Xia, Kai Yi, Bibhuprasad Mahakud, Rajat Gupta, Lei Feng, Shuai Liu, Davide Cieri, Aram Avetisyan, Charles Dietz, Alexandre Aubin, Michal Simon, Matteo Marone, Amr Mohamed, Juan Ramon Castiñeiras De Saa