有些数学问题,看似与三角形毫无联系,但倘若充分地挖掘题设中的内涵,便可便问题与某个特定的三角形紧密相联。从而便得到一个别出心裁的解法。一般而言,我们可据题设从构成一个三角形的角与边之条件来设计相应的模型三角形。例1 设α、β为锐角,且3sin2α-2sin2β=0,3cos2α+2cos2β-3=0,求证α+2β=π/2 分析:由Scos2α+2cos2β-3=0可知2α、2β均应为锐角,变换题设两等式有 3sin2α=2sin2β,3cos2α+2cos2β=3。 Some mathematical problems seem to have nothing to do with triangles, but if you fully explore the connotations of the questions, the problems can be closely related to a particular triangle. So you get an ingenious solution. In general, we can design the corresponding model triangles from the conditions that form the corners and edges of a triangle. Example 1 Let α and β be acute angles, and 3sin2α-2sin2β=0, 3cos2α+2cos2β-3=0, verify α+2β=π/2 Analysis: From Scos2α+2cos2β-3=0, we know that 2α and 2β should be acute angles. There are two equations for the transformation problem: 3sin2α=2sin2β and 3cos2α+2cos2β=3.