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在三角函数的学习过程中,我发现除了一些特殊角的三角函数值可直接计算之外,还有一些非特殊角组成的三角函数式,可通过三角变换“整体”地求出它们的值.如cos20°cos40°cos80°,就可以巧妙地应用二倍角公式转化为1/8sin20°·8sin20°cos20°cos40°cos80°=1/8sin20°·sin160=1/8.实际上,与之形式相同的,如cos40°cos80°cos160°也同样可求得. 在做完这道题后,出于爱好研究一个系列问题的习惯,我又思考:cos40°+cos80°+cos160°的值易求吗?我先用计算器算,发现其值为0.再用理论进行推导.考虑到这些角与特殊角60°的差异,不难转化为cos(60°-20°)+cos(60°+20°)-cos20°,展开得其值为0.做完这道题后,又使我联想到求cos20°+cos40°+cos80°的值的问题.
In the course of learning the trigonometric function, I found that trigonometric functions other than special angles can be directly calculated in addition to the trigonometric function values of some special angles. They can be obtained by trigonometric transformation. Values such as cos20°cos40°cos80°, can be applied skillfully to convert the double-angle formula into 1/8sin20°·8sin20°cos20°cos40°cos80°=1/8sin20°·sin160=1/8. Actually, with The same form, such as cos40°cos80°cos160° can also be obtained. After doing this problem, because of the habit of studying a series of problems, I also think: the value of cos40°+cos80°+cos160° Do you want? I used a calculator to calculate that it was 0. Then use the theory to derive. Taking into account the difference between these angles and special angle 60 °, it is not difficult to convert to cos (60 ° -20 °) + cos (60 °+20°)-cos20°, expanded to a value of 0. After finishing this question, it also reminded me of the problem of finding the value of cos20°+cos40°+cos80°.