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解三角方程,用不同的方法求解,所求出的解集的表示式往往各异。例如解sin4x+sin2x=0; 化为2sin3xcosx=0,得解集化为sin2x(2cos2x+1)=0,得解集高一课本指出,不同的解法,虽然得到的解集表示形式不同,但实质是相等的。但课文没有证明。《数学教学》90年第六期《三角方程解集的等价性判别》一文中发表了一种判别法,拜读后,很受启发。在该文的基础上,现提出判别三角方程解集等价的一种方法——“相同余数法”,作为对该文的完善。由于三角函数是周期函数,所以三角方程的解有无限多个,而且有一定规律,解集是由一个或几个“双向等差数列”的各项所组成(注:等差数列n取自然数,而解集数列n取整数),如
Solve triangular equations and solve them with different methods. The expressions of the solution sets obtained are often different. For example, the solution sin4x + sin2x = 0; into 2sin3xcosx = 0, the solution set is sin2x (2cos2x + 1) = 0, the solution set high school textbooks pointed out that different solutions, although the solution set representation obtained is different, but The essence is equal. However, there is no proof of the text. In Mathematical Teaching in the sixth issue of “Trigonometric Equations Solution Equivalence Discrimination,” published in the article, a discriminating method was published. After reading it, he was inspired. On the basis of this paper, a method of discriminating the equivalent set of trigonometric equations is proposed, the “identical remainder method”, as a refinement of this paper. Since the trigonometric function is a periodic function, the solution of the trigonometric equation has infinitely many, and there are certain rules. The solution set consists of one or several “bidirectional arithmetic series” (Note: The arithmetic series n takes natural numbers. , and the solution set number n takes an integer, such as