You know, cosine is adjacent over hypotenuse.

你们知道 cos邻边除以斜边。

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

我懂了，正弦对边比斜边。。

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

斜边平方等于直边平方和可把我给惊呆了。

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

几何中最著名关系直三边和斜边之间关系。

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（勾股）做出了精彩证明；这个说，对于一个直三，一边平方加上另一边平方等于斜边平方。

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分量应该等於乘以斜边所以同样 这里x分量。

Root 2 simply is the hypotenuse of a right triangle

而 √2 只两边长度为 1 直三之斜边。

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

取四个全等直三，两边分别长 a 和 b，斜边长 c。

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

直三斜边多少 直三斜边多少。

Arrange them so that their hypotenuses form a tilted square.

将它们排列起来，使其斜边成一个倾斜正方。

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

我想他大概五岁就知道什么斜边了。

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

好吧，它正沿着斜边，沿着一个倾斜度慢慢前行。

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.

他们都为着名勾股提出了简洁证明，该指出， 对于一个直三，一条直边平方加上另一条直边平方等于斜边平方。

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， 古代历史学家描述就，他们实际上试图画一个直三， 两个入口分别位于斜边两端。

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， 我们可以画一个小直三，其中球面矩高度斜边，其投影其中一个直边。

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平方81， 再加上81，因此斜边长度就162平方根， 大约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 相对边 x 方向向量 A，斜边向量 A。这意味着 x 方向向量 A 等于 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.

也就说, 在这种情况下, 斜边和边之间关系不能用任何数值比率表示, 只能用无数表示。