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基本触发身份
Transcript
Fundamental trig identities. So far we've talked about the three main trig functions, sine, cosine, and tangent.
这三个是比例。但从技术上讲，从萨哈塔拉三角的三面，
实际上可以创造六个比率。因此，六个中的每一个都是单独的触发功能。
真的，所有六个都很重要。所以我们已经知道了三个。 So let's take a look at this SOHCAHTOA triangle. There's our familiar SOHCAHTOA triangle, happens to have an angle of 41 degrees. There's an opposite, adjacent, hypotenuse side. And so certainly three of the ratios we can create are the familiar soh cah toa比率。
But there are three more ratios we can create. And here they are. Cotangent与对面相邻。近距离斜边。 And cosecant is hypotenuse over opposite opposite. And those are the six ratios altogether.
So wait a second. What are those names? Let's look at these names very carefully. Here are the full names. So we've already talked about sine, cosine and tangent. And now we're talking about cotangent, secant and cosecant.
And notice the way they're listed here. If you can remember the three on the left the three on the right is just the same name with co in front of it. So at least some of these names have their origin in geometric relationships. Let's talk about this for a minute. So now let's look at a circle.
可以是单位圈，半径为1，中心在原点。因此AB和CD平行于Y轴。 So we have two vertical segments there, AB and CD. And it looks like B is the point where that radius line intersects the circle, it continues on. And D looks like it's tangent to the circle where it crosses the x-axis.
好的，所以注意一些事情。即，在三角形oab中，圆圈内的三角形，ob，半径为1。 当然，OA是余弦，AB是正弦。所以这是熟悉的Sohcahtoa比率。 现在，看三角形，略大三角形，强迫。所以这个是那个来的，从o开始， passes through B all the way out to C.
Drops down to D and goes back along the X axis. Well, in that triangle, OD is 1. 因此，这意味着相反的CD超过1等于切线。所以切线等于CD。 这意味着相邻的斜边OC超过1，是割线的。
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. 它通过圆圈并在一点触摸它。事实上，这是一个切线。
注意,OC,秒ant, actually cuts through the circle. And so this is what's known in geometry as a secant line. 这就是为什么这两个函数具有那些名称，因为一个代表切线段的长度。 And one represents the length of a secant segment. And so, if you're a very visual person, 这可能会帮助您记住这些事情一点点。
Okay, sine and cosine are the most elementary trig functions. And we can actually express the other four in terms of them. And these are really important formulas to know. Tangent we can write as sine over cosine. Cotangent we can write as cosine over sine. So notice that those two are reciprocals.
切线和cotangent是互惠的。云是余弦和用杂项的倒数是正弦的倒数。 Notice that people get confused sometimes cuz they think that the s and the s should go together. The c and the c should go together, they don't. Secant is the reciprocal of cosine.
求生是正弦的倒数。所以测试可能会给你一个，如果你需要它一个问题。 但它可能希望你也记住它。所以有这四个记忆真的很好。 Now in the first lesson on trig, we mentioned the fundamental Pythagorean identity, cosine squared plus sine squared equals one.
Now that we have two more functions, we can also express the other Pythagorean identities. One of them is tangent squared plus 1 equals secant squared. One of them is cotangent squared plus 1 equals cosecant squared. 因此，如果需要问题，测试很可能会给您这些等式。
但他们可以作为确认答案的快捷方式或一种方法。我会说的另一件事是，如果你计划采取微积分我保证， 一旦您在微积分中，我绝对保证您需要了解所有三个方程式。 所以我会说一些关于这些事情的事情。当然，您可以盲目地记住它们，但我们不建议。
我们真正推荐的是理解它们。等等，如果你从顶部的那个开始，余弦方形加上正弦方 equals 1, you could divide everything on both sides by cosine squared. You'd get the top Pythagorean identity at the bottom tangent and secant. Or you could divide everything in cosine squared plus sine squared equals 1 by sine squared.
然后你得到底部的一个，cotangent平方和腐烂。或者，您可以通过ABC和ABC返回原始的Sohcahtoa三角形 从Pythagorean定理开始。平方加B平方等于C平方。 您可能会记住，我们得到了这个顶级的毕达哥兰身份，余弦方形加上正弦方形等于1。
We got that from taking a squared plus b squared plus c squared and dividing everything, all three terms, by c squared. 嗯，而不是通过C分开，我们可以通过平方或B平方分开所有三个术语。 如果您这样做，然后从比率分发，您将产生这两个毕达哥拉斯的身份。
And so I strongly suggest, do that on your own. Show in a couple different ways that you can come up with all these equations, because then you'll really understand them. Okay, now we can move on to 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 o the following is the value of tangent theta?
All right, well let's think about this. We have two sides there, we're given b and c. And, of course, c is the hypotenuse. b is the opposite, and tangent is opposite over adjacent. 我们有相反的，我们没有邻近的。所以我们需要第三方。
好吧，我们可以使用毕达哥拉斯定理。因此，毕达哥兰定理告诉我们B个平方加上相邻的一侧 平方等于C平方。我们可以用相邻的一面解决这个问题。 Adjacent squared equals c squared minus b squared. Take a square root of both sides.
Notice that taking a square root, we cannot take a square root of c and b separately. We have to leave it as that expression, c squared minus b squared. But that is an expression for the length of the adjacent side. c squared minus b squared. Well, now we're golden because tangent is opposite over adjacent.
我们有相反的，我们有邻近的。在邻近和 that would equal b over the square root of c squared minus b squared. And in fact that is answer C. We go back to the problem and we choose answer C. In summary, we introduced the other three trig functions, cotangent, secant, and cosecant.
We discussed how to express the other four in terms of sine and cosine. So it's very good to understand how they fit into the SOHCAHTOA triangle. It's very good to understand how they're related to sine and cosine. And finally we discussed the three Pythagorean Identities.
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真的，所有六个都很重要。所以我们已经知道了三个。 So let's take a look at this SOHCAHTOA triangle. There's our familiar SOHCAHTOA triangle, happens to have an angle of 41 degrees. There's an opposite, adjacent, hypotenuse side. And so certainly three of the ratios we can create are the familiar soh cah toa比率。
But there are three more ratios we can create. And here they are. Cotangent与对面相邻。近距离斜边。 And cosecant is hypotenuse over opposite opposite. And those are the six ratios altogether.
So wait a second. What are those names? Let's look at these names very carefully. Here are the full names. So we've already talked about sine, cosine and tangent. And now we're talking about cotangent, secant and cosecant.
And notice the way they're listed here. If you can remember the three on the left the three on the right is just the same name with co in front of it. So at least some of these names have their origin in geometric relationships. Let's talk about this for a minute. So now let's look at a circle.
可以是单位圈，半径为1，中心在原点。因此AB和CD平行于Y轴。 So we have two vertical segments there, AB and CD. And it looks like B is the point where that radius line intersects the circle, it continues on. And D looks like it's tangent to the circle where it crosses the x-axis.
好的，所以注意一些事情。即，在三角形oab中，圆圈内的三角形，ob，半径为1。 当然，OA是余弦，AB是正弦。所以这是熟悉的Sohcahtoa比率。 现在，看三角形，略大三角形，强迫。所以这个是那个来的，从o开始， passes through B all the way out to C.
Drops down to D and goes back along the X axis. Well, in that triangle, OD is 1. 因此，这意味着相反的CD超过1等于切线。所以切线等于CD。 这意味着相邻的斜边OC超过1，是割线的。
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. 它通过圆圈并在一点触摸它。事实上，这是一个切线。
注意,OC,秒ant, actually cuts through the circle. And so this is what's known in geometry as a secant line. 这就是为什么这两个函数具有那些名称，因为一个代表切线段的长度。 And one represents the length of a secant segment. And so, if you're a very visual person, 这可能会帮助您记住这些事情一点点。
Okay, sine and cosine are the most elementary trig functions. And we can actually express the other four in terms of them. And these are really important formulas to know. Tangent we can write as sine over cosine. Cotangent we can write as cosine over sine. So notice that those two are reciprocals.
切线和cotangent是互惠的。云是余弦和用杂项的倒数是正弦的倒数。 Notice that people get confused sometimes cuz they think that the s and the s should go together. The c and the c should go together, they don't. Secant is the reciprocal of cosine.
求生是正弦的倒数。所以测试可能会给你一个，如果你需要它一个问题。 但它可能希望你也记住它。所以有这四个记忆真的很好。 Now in the first lesson on trig, we mentioned the fundamental Pythagorean identity, cosine squared plus sine squared equals one.
Now that we have two more functions, we can also express the other Pythagorean identities. One of them is tangent squared plus 1 equals secant squared. One of them is cotangent squared plus 1 equals cosecant squared. 因此，如果需要问题，测试很可能会给您这些等式。
但他们可以作为确认答案的快捷方式或一种方法。我会说的另一件事是，如果你计划采取微积分我保证， 一旦您在微积分中，我绝对保证您需要了解所有三个方程式。 所以我会说一些关于这些事情的事情。当然，您可以盲目地记住它们，但我们不建议。
我们真正推荐的是理解它们。等等，如果你从顶部的那个开始，余弦方形加上正弦方 equals 1, you could divide everything on both sides by cosine squared. You'd get the top Pythagorean identity at the bottom tangent and secant. Or you could divide everything in cosine squared plus sine squared equals 1 by sine squared.
然后你得到底部的一个，cotangent平方和腐烂。或者，您可以通过ABC和ABC返回原始的Sohcahtoa三角形 从Pythagorean定理开始。平方加B平方等于C平方。 您可能会记住，我们得到了这个顶级的毕达哥兰身份，余弦方形加上正弦方形等于1。
We got that from taking a squared plus b squared plus c squared and dividing everything, all three terms, by c squared. 嗯，而不是通过C分开，我们可以通过平方或B平方分开所有三个术语。 如果您这样做，然后从比率分发，您将产生这两个毕达哥拉斯的身份。
And so I strongly suggest, do that on your own. Show in a couple different ways that you can come up with all these equations, because then you'll really understand them. Okay, now we can move on to 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 o the following is the value of tangent theta?
All right, well let's think about this. We have two sides there, we're given b and c. And, of course, c is the hypotenuse. b is the opposite, and tangent is opposite over adjacent. 我们有相反的，我们没有邻近的。所以我们需要第三方。
好吧，我们可以使用毕达哥拉斯定理。因此，毕达哥兰定理告诉我们B个平方加上相邻的一侧 平方等于C平方。我们可以用相邻的一面解决这个问题。 Adjacent squared equals c squared minus b squared. Take a square root of both sides.
Notice that taking a square root, we cannot take a square root of c and b separately. We have to leave it as that expression, c squared minus b squared. But that is an expression for the length of the adjacent side. c squared minus b squared. Well, now we're golden because tangent is opposite over adjacent.
我们有相反的，我们有邻近的。在邻近和 that would equal b over the square root of c squared minus b squared. And in fact that is answer C. We go back to the problem and we choose answer C. In summary, we introduced the other three trig functions, cotangent, secant, and cosecant.
We discussed how to express the other four in terms of sine and cosine. So it's very good to understand how they fit into the SOHCAHTOA triangle. It's very good to understand how they're related to sine and cosine. And finally we discussed the three Pythagorean Identities.
