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一个三角不定方程的机器解法

徐嘉1, 姚勇2   

  1. 1. 西南民族大学计算机科学与技术学院, 成都 610041; 2. 中国科学院成都计算机应用研究所, 成都  610041
  • 收稿日期:2010-11-04 出版日期:2011-07-25 发布日期:2011-09-27

徐嘉, 姚勇. 一个三角不定方程的机器解法[J]. 系统科学与数学, 2011, 31(7): 786-793.

XU Jia, YAO Jia. AUTOMATED SOLVING METHODS FOR A TRIGONOMETRIC INDETERMINATE EQUATION[J]. Journal of Systems Science and Mathematical Sciences, 2011, 31(7): 786-793.

AUTOMATED SOLVING METHODS FOR A TRIGONOMETRIC INDETERMINATE EQUATION

XU Jia1, YAO Jia2   

  1. 1. College of Computer Science and Technology, Southwest University for Nationalities,   Chengdu 610041; 2. Chengdu Institute of Computer Applications, Chinese Academy of Sciences, Chengdu 610041
  • Received:2010-11-04 Online:2011-07-25 Published:2011-09-27
研究了如下具有几何意义的三角不定方程(采用角度制)$$\frac{\sin(x^\circ)\sin(y^\circ)\sin(z^\circ)}{\sin(A^\circ-x^\circ)\sin(B^\circ-y^\circ)\sin(C^\circ-z^\circ)}=1,$$
其中$A,B,C$是给定的正整数,满足$2\leq A\leq B\leq C$且$A+B+C=180$;\ $1\leq x\leq A-1,\ 1\leq y\leq B-1,\ 1\leq z\leq C-1$.设计了两种求解方案, 使用计算机按两种求解方案完整的求出了上述方程的所有正整数解.
This paper studies the following trigonometric indeterminate equation with geometric significance~(using degree measure) $$\frac{\sin(x^\circ)\sin(y^\circ)\sin(z^\circ)}{\sin(A^\circ-x^\circ)\sin(B^\circ-y^\circ)\sin(C^\circ-z^\circ)}=1,$$ where $A,B,C$ are given positive integers satisfying $2\leq A\leq B\leq C,
 A+B+C=180$,\ $1\leq x\leq A-1, \ 1\leq y\leq B-1,\ 1\leq z\leq C-1$. Two methods to the above equation are formulated. Following these two methods,
all positive integer solutions of the equation are obtained via computer.

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