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

我懂了，正弦是对边比斜边。是的。

You know, cosine is adjacent over hypotenuse.

你们知道 cos是邻边除以斜边。

Root 2 simply is the hypotenuse of a right triangle

而 √2 只是两边长度为 1 的 直角三角形之斜边。

Arrange them so that their hypotenuses form a tilted square.

将它们排列起，斜边形成一个斜的正方形。

Opposite over hypotenuse is equal to sine of an angle.

斜边分之对边 的值即是 sin 函数的值。

So you have the opposite over the hypotenuse.

这里是对边比斜边。

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，两边分别长 a 和 b，斜边长 c。

And this comes straight out of this magnitude, which is opposite to the angle over the hypotenuse.

这是因为这个向量，让它的量值当分母。

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边和斜边之间的关系。

What involves the hypotenuse and the opposite side?

什么涉及斜边和对边？

So what involves the opposite and the hypotenuse?

那么什么涉及对立面和斜边呢？

Sine theta equals opposite over hypotenuse.

正弦θ等于对边除以斜边。

Cosine theta equals adjacent over hypotenuse.

cosine theta 等于相邻在斜边上。

It's adjacent over hypotenuse. And that's square root of 3 over 2.

它在斜边上相邻。那是 3 除 2 的平方根。

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

斜边平方等于直角边平方和可把我给惊呆了。

That states that the squares of the sides of a right triangle add up to the square of its hypotenuse.

这表明直角三角形各边的平方加起等于斜边的平方。

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, 斜边和边之间的关系不能用任何数值比率表示, 只能用无理数表示。

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 相对的边是 x 方向的向量 A，斜边是向量 A。这意味着 x 方向的向量 A 等于 A 正弦 theta。

The side  adjacent theta is vector A in the y-direction, and the hypotenuse is still vector A. That means  vector A in the y-direction equals A cosine theta.

相邻的θ边是y方向的向量A, 斜边仍然是向量A, 这意味着y方向的向量A等于余弦θ。

We want to know how much the height of our sphere-rectangle gets squished during this projection, which is the ratio of this hypotenuse to the leg on the right.

我们想知道在投影过程中球体矩形的高度被压扁了多少，即斜边与右侧边的比率。