You know, cosine is adjacent over hypotenuse.

你们知道 cos是邻除以。

The most famous relation in geometry is the relation between the sides and the hypotenuse of a right triangle.

几何中最著名的关系是直角三角形的和之间的关系。

I get it! Sin is opposite hypotenuse... Yes.

，正弦是对比。是的。

You know, I was blown away that the square of the hypotenuse was the sum of the squares of the opposite sides.

平方等直角平方和可把给惊呆。

They all came up with elegant proofs for the famous Pythagorean theorem; the rule that says for a right triangle, the square of one side plus the square of the other side is equal to the square of the hypotenuse.

他们都对达哥拉斯定理（勾股定理）做出精彩的证明；这个定理是说，对一个直角三角形，一的平方加上另一的平方等的平方。

That the x component is going to be the cosine of the angle between the hypotenuse and the x component times the hypotenuse.

这个x分量应该等於乘以所以同样的 这里的x分量。

Root 2 simply is the hypotenuse of a right triangle

而 √2 只是两长度为 1 的 直角三角形之。

Take four identical right triangles with side lengths a and b and hypotenuse length c.

取四个全等的直角三角形，两分别长 a 和 b，长 c。

So the magnitude of this over the hypotenuse of this right triangle.

直角三角形的是多少 直角三角形的是多少。

Arrange them so that their hypotenuses form a tilted square.

将它们排列起来，使其形成一个倾的正方形。

I think he knew what a hypotenuse was when he was about five.

想他大概五岁就知道什么是。

Well, it's ambling its way along a hypotenuse, along a slopey angle.

好吧，它正沿着，沿着一个倾的角度慢慢前行。

They all came up with elegant proofs for the famous Pythagorean theorem, the rule that says for a right triangle, the square of one side plus the square of the other side is equal to the square of the hypotenuse.

他们都为着名的勾股定理提出简洁的证明，该定理指出， 对一个直角三角形，一条直角的平方加上另一条直角的平方等的平方。

And so what they thought, and this is what the ancient historians described, is that they effectively tried to draw a right angle triangle with the two entrances at each end of the hypotenuse.

他们认为， 古代历史学家描述的就是，他们实际上试图画一个直角三角形， 两个入口分别位的两端。

To think about the projection, let's make a little right triangle like this, where what was the height of our spherical rectangle is the hypotenuse, and its projection is one of the legs.

要想考虑投影， 们可以画一个小直角三角形，其中球面矩形的高度是，其投影是其中一个直角。

So 9 squared, which is 81 plus 81, so to figure out what that hypotenuse is, it is the square root of 162 or it is around 12.7.

所以9的平方是81， 再加上81，因此的长度就是162的平方根， 大约是12.7。

The side opposite theta is vector  A in the x-direction, and on the hypotenuse is vector A. That means vector A in the  x-direction equals A sine theta.

与 theta 相对的是 x 方向的向量 A，是向量 A。这意味着 x 方向的向量 A 等 A 正弦 theta。

That is, in such cases the relation between hypotenuse and sides cannot be expressed by any numerical ratio, only by an irrational number.

也就是说, 在这种情况下, 和之间的关系不能用任何数值比率表示, 只能用无理数表示。