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Fundamental Trig Identities
成绩单
基本三角恒等式。到目前为止，我们已经讨论了三个主要的三角函数，正弦、余弦和切线。
Those three are ratios but technically from the three sides of the SOHCAHTOA triangle, it's actually possibly to create six ratios.
And so each of the six is a separate trig function, and really all six are important to know.
我们已经认识三个了。那么让我们来看看SOHCAHTOA三角形。 这是我们熟悉的Sohcahto三角形，它恰好有41度角，有一个相对的斜边。 当然，我们能创造的三个比率是我们熟悉的SOHCAHTOA比率，但我们还能创造另外三个比率。
在这里，余切是相邻的，正割是斜边，余割是斜边， and those are the six ratios all together. So wait a second, what are those names? 让我们仔细看看这些名字，这里是全名。我们已经讨论过正弦，余弦和切线。
现在我们讨论的是余切，割线和余切。注意这里的列表，如果你记得左边的三个， the three on the right, it's just the same name with co in front of it. So at least some of these names have their origin in geometric relationships. 让我们谈一下这个。现在让我们看一个圆，可能是单位圆。
半径为1，中心位于原点。所以AB和CD平行于y轴。 所以我们有两个垂直的部分，AB和CD。看起来B是半径线与圆相交的点， it continues on, and D looks like it's tangent to the circle where it crosses the x-axis.
Okay, so notice a few things, that in triangle OAB the triangle inside the circle OB, the radius is 1, 当然OA是余弦，AB是正弦，好吗？所以这就是我们熟悉的SOHCAHTOA比率。 现在看看大一点的三角形强迫症。这个从O开始，经过B，一直到C， drops down to D, and comes back along the x-axis.
Well, in that triangle, OD is 1. And so that would mean that opposite CD over 1 equals the tangent, so the tangent equals CD. And it means that hypotenuse over adjacent OC over 1 is secant, so OC equals the secant. But here's the really cool thing about this diagram.
Notice that CD, the segment that has a length equal to the tangent is actually tangent to the circle. It passes the circle and touches it at one point, that is in fact a tangent line. Notice that OC, which is the secant, actually cuts through the circle, and so this is what's known in geometry as a secant line. And so that's why those two functions have those names, because one represents 切线段的长度，其中一个表示正割段的长度。
所以如果你是一个非常视觉的人,可能help you remember these things a little bit. 正弦和余弦是最基本的三角函数，我们可以用它们来表示其他四个函数 这些都是非常重要的公式。切线可以写成余弦的符号。
余切我们可以写成余弦对正弦，注意这两个是倒数，正切和余切是倒数。 割线是余弦的倒数，余割是正弦的倒数。注意，人们有时会感到困惑，因为他们认为 s应该一起走，c和c应该一起走，它们不是。割线是余弦的倒数。
余割是正弦的倒数。因此，如果你在问题中需要，测试可能会给你其中一个，但它可能会 expect you to remember it as well, so it's really good to have those four memorized. Now on the first lesson on training, we mentioned the fundamental Pythagorean 恒等式，余弦平方加正弦平方等于1。现在我们还有两个功能， we can also express the other Pythagorean identities.
其中一个是切线平方加1等于正割平方。其中一个是余切平方加1等于余切平方。 So the test quite likely would give you these equations if a problem required them, but they may serve as a shortcut or a way to confirm the answer. 我要说的另一件事是，如果你打算学微积分，我保证，我绝对保证， that you need to know all three of these equations cold once you're in calculus.
So I'll say a few things about these. Of course you can blindly memorize them, but we don't recommend that, 我们真正建议的是理解它们。如果你从最上面的开始，余弦平方加正弦平方 等于1，你可以把两边的东西除以余弦的平方，你会得到顶部的毕达哥拉斯恒等式，在底部的切线和正割处，或者 你可以用余弦平方加上正弦平方等于1除以正弦平方，最后得到余切平方和余切平方。
Ultimately, you could go back to the original SOHCAHTOA triangle with ABC and start with the Pythagorean theorem, A squared plus B squared equals C squared. 你可能还记得，我们得到了这个最高的毕达哥拉斯恒等式，余弦的平方加上正弦的平方等于1，我们得到了这个恒等式，取一个平方 加上B的平方加上C的平方，把所有三项除以C的平方。假设除以C的平方， we could divide all three terms by either A squared or B squared.
如果你这样做，然后从比率中，再细分，三角函数是什么，你会得到这两个毕达哥拉斯恒等式。 所以我强烈建议你自己去做。用两种不同的方法证明你能得出所有这些方程， because then you'll really understand them. Okay, now we can move onto a practice problem.
Pause the video and we'll talk about this. All right, in the triangle to the right, in terms of b and c, which of the following is the value of tangent theta? All right, well let's think about this. 我们有两个边，分别是b和c，当然c是斜边，b是对边，还有 tangent is opposite over adjacent.
我们有相反的，我们没有相邻的，所以我们需要第三面。 Well, we can use the Pythagorean theorem. So the Pythagorean theorem tells us that b squared plus whatever the adjacent side 平方等于c的平方，我们可以用相邻的符号来求解。相邻的平方等于c的平方减去b的平方，取两边的平方根。
注意，取一个平方根，我们不能分别取c和b的平方根，我们必须把它留在这个表达式上，c的平方减去b的平方。 But that is an expression for the length of the adjacent sides, c squared minus b squared. 好吧，现在我们是金色的，因为切线和相邻的相反。我们有相反的，我们有相邻的。
So, opposite over adjacent, and that would equal b over the square root of c squared minus b squared, and in fact that is answer c. 我们回到问题，选择答案c。总之，我们介绍了其他三种三角函数，余切函数，割线函数， 和余割。我们讨论了如何用正弦和余弦表示其他四个，所以 很好地理解了它们是如何适合于SOHCAHTOA三角形的，很好地理解了它们是如何与正弦和余弦相关的。
And finally, we discussed the three Pythagorean Identities.
阅读完整的成绩单
我们已经认识三个了。那么让我们来看看SOHCAHTOA三角形。 这是我们熟悉的Sohcahto三角形，它恰好有41度角，有一个相对的斜边。 当然，我们能创造的三个比率是我们熟悉的SOHCAHTOA比率，但我们还能创造另外三个比率。
在这里，余切是相邻的，正割是斜边，余割是斜边， and those are the six ratios all together. So wait a second, what are those names? 让我们仔细看看这些名字，这里是全名。我们已经讨论过正弦，余弦和切线。
现在我们讨论的是余切，割线和余切。注意这里的列表，如果你记得左边的三个， the three on the right, it's just the same name with co in front of it. So at least some of these names have their origin in geometric relationships. 让我们谈一下这个。现在让我们看一个圆，可能是单位圆。
半径为1，中心位于原点。所以AB和CD平行于y轴。 所以我们有两个垂直的部分，AB和CD。看起来B是半径线与圆相交的点， it continues on, and D looks like it's tangent to the circle where it crosses the x-axis.
Okay, so notice a few things, that in triangle OAB the triangle inside the circle OB, the radius is 1, 当然OA是余弦，AB是正弦，好吗？所以这就是我们熟悉的SOHCAHTOA比率。 现在看看大一点的三角形强迫症。这个从O开始，经过B，一直到C， drops down to D, and comes back along the x-axis.
Well, in that triangle, OD is 1. And so that would mean that opposite CD over 1 equals the tangent, so the tangent equals CD. And it means that hypotenuse over adjacent OC over 1 is secant, so OC equals the secant. But here's the really cool thing about this diagram.
Notice that CD, the segment that has a length equal to the tangent is actually tangent to the circle. It passes the circle and touches it at one point, that is in fact a tangent line. Notice that OC, which is the secant, actually cuts through the circle, and so this is what's known in geometry as a secant line. And so that's why those two functions have those names, because one represents 切线段的长度，其中一个表示正割段的长度。
所以如果你是一个非常视觉的人,可能help you remember these things a little bit. 正弦和余弦是最基本的三角函数，我们可以用它们来表示其他四个函数 这些都是非常重要的公式。切线可以写成余弦的符号。
余切我们可以写成余弦对正弦，注意这两个是倒数，正切和余切是倒数。 割线是余弦的倒数，余割是正弦的倒数。注意，人们有时会感到困惑，因为他们认为 s应该一起走，c和c应该一起走，它们不是。割线是余弦的倒数。
余割是正弦的倒数。因此，如果你在问题中需要，测试可能会给你其中一个，但它可能会 expect you to remember it as well, so it's really good to have those four memorized. Now on the first lesson on training, we mentioned the fundamental Pythagorean 恒等式，余弦平方加正弦平方等于1。现在我们还有两个功能， we can also express the other Pythagorean identities.
其中一个是切线平方加1等于正割平方。其中一个是余切平方加1等于余切平方。 So the test quite likely would give you these equations if a problem required them, but they may serve as a shortcut or a way to confirm the answer. 我要说的另一件事是，如果你打算学微积分，我保证，我绝对保证， that you need to know all three of these equations cold once you're in calculus.
So I'll say a few things about these. Of course you can blindly memorize them, but we don't recommend that, 我们真正建议的是理解它们。如果你从最上面的开始，余弦平方加正弦平方 等于1，你可以把两边的东西除以余弦的平方，你会得到顶部的毕达哥拉斯恒等式，在底部的切线和正割处，或者 你可以用余弦平方加上正弦平方等于1除以正弦平方，最后得到余切平方和余切平方。
Ultimately, you could go back to the original SOHCAHTOA triangle with ABC and start with the Pythagorean theorem, A squared plus B squared equals C squared. 你可能还记得，我们得到了这个最高的毕达哥拉斯恒等式，余弦的平方加上正弦的平方等于1，我们得到了这个恒等式，取一个平方 加上B的平方加上C的平方，把所有三项除以C的平方。假设除以C的平方， we could divide all three terms by either A squared or B squared.
如果你这样做，然后从比率中，再细分，三角函数是什么，你会得到这两个毕达哥拉斯恒等式。 所以我强烈建议你自己去做。用两种不同的方法证明你能得出所有这些方程， because then you'll really understand them. Okay, now we can move onto a practice problem.
Pause the video and we'll talk about this. All right, in the triangle to the right, in terms of b and c, which of the following is the value of tangent theta? All right, well let's think about this. 我们有两个边，分别是b和c，当然c是斜边，b是对边，还有 tangent is opposite over adjacent.
我们有相反的，我们没有相邻的，所以我们需要第三面。 Well, we can use the Pythagorean theorem. So the Pythagorean theorem tells us that b squared plus whatever the adjacent side 平方等于c的平方，我们可以用相邻的符号来求解。相邻的平方等于c的平方减去b的平方，取两边的平方根。
注意，取一个平方根，我们不能分别取c和b的平方根，我们必须把它留在这个表达式上，c的平方减去b的平方。 But that is an expression for the length of the adjacent sides, c squared minus b squared. 好吧，现在我们是金色的，因为切线和相邻的相反。我们有相反的，我们有相邻的。
So, opposite over adjacent, and that would equal b over the square root of c squared minus b squared, and in fact that is answer c. 我们回到问题，选择答案c。总之，我们介绍了其他三种三角函数，余切函数，割线函数， 和余割。我们讨论了如何用正弦和余弦表示其他四个，所以 很好地理解了它们是如何适合于SOHCAHTOA三角形的，很好地理解了它们是如何与正弦和余弦相关的。
And finally, we discussed the three Pythagorean Identities.
