三种方法都可以利用直角三角形的性质及代数计算来加以证明。

Three methods can be proved using the feature of a right triangle and algebraic computation.

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虽然这些计算无非是最基本的三角形和代数计算，但是所使用的公式非常复杂。

Although these calculations are nothing more than basic trigonometry and algebra, the formulas are very complicated.

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不幸的是，这一方法仍然包含了一系列复杂的代数计算，还涉及了复杂得三角函数。

Unfortunately, the solution still included a complicated set of sums of algebraic series involving tricky powers of trigonometric functions.

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Now, we get to draw some lessons out of this thing, so everybody who's feeling a little bit shell shocked from having been doing algebra and calculus and drawing pictures and feeling like they've been cheated into taking a class that looks far too much like economics, calm down we're going to actually talk right now.

下面我们从中总结点经验出来,那些因为代数和微积分计算还有绘图,而感到十分不爽的同学,你们是不是感觉被我忽悠了,才会选这门一点都不像经济学的课啊,稍安勿躁,我们马上切入正题

耶鲁公开课 - 博弈论课程节选

Now, unlike high school math or in algebra Z where you call things X and Y and Z, in programming, in computer science, you're actually dealing with humans where it's useful to have a variable name that's more descriptive than X and Y and Z.

不像高中数学或者代数中,称为X和Y和，在程序设计和计算机科学里，你实际上是在和人打交道,在这里有个描述性比xyz更强的,变量名称是很重要的。

哈佛公开课 - 计算机科学课程节选