一次习题课上我出了这样一道选择题让同学们做: 方程arcsinx+arccosy=π所代表的曲线是下面四个图形中的( ) 显然上述四个结论中正确的结论有且只有一个。然而令人吃惊的是,同学们的答案中,四个结论竟无一不被选中。为查明错因所在,我分别抽出选中(A)、(B)、(C)、(D)的各一人,让他们将自己的判断过程写在黑板上。学生甲:将方程arcsimx+arccosy=π两边取正弦得xy+1-x~2(1/2)·1-y~2(1/2)=0,移项得1-x~2(1/2)·1-y~2(1/2)=-xy (1)平方化简:x~2+y~2=1 (2) 由(2)知原方程所代表的曲线为 (A) 学生乙:将方程arcsinx+arccosy=π In an exercise class I gave a multiple-choice question for classmates to do: The curve represented by arcsinx+arccosy=π is () in the following four graphs. Obviously, there are only one correct conclusion among the above four conclusions. What was surprising, however, was that none of the four conclusions of the students’ answers was selected. In order to find out the cause of the error, I took out each person selected (A), (B), (C) and (D) and asked them to write their own judgments on the blackboard. Student A: Take the equation arcsx+arccosy=π to get sine xy+1-x~2(1/2)·1-y~2(1/2)=0 and shift the term to 1-x~2(1) /2)·1-y~2(1/2)=-xy (1) Simplification of square: x~2+y~2=1 (2) The curve represented by (2) the original equation is (A) Student B: The equation arcsinx+arccosy=π