Back to Table of Contents
介绍Sohcahtoa.
Transcript
三角学，索哈托介绍。本课程假设您熟悉类似三角形的想法，
covered in the GEOMETRY module. If the idea of similar triangles is absolutely unfamiliar to you,
it might be helpful to watch that video in the geometry module, before watching the trigonometry videos.
回想一下，如果我们在一个三角形中只知道两个角度等于另一个三角形的两个角度，那么两个三角形必须是类似的 that means that they have the same basic shape. One is just a scaled up, or a scaled down version of the other. 所有三个角度都是相同的，它是相同的基本形状。一旦我们知道这两个三角形是相似的， we know that all their sides are proportional.
It's very easy to show that two triangles are similar, and once we know that, we get a lot of information. 所有三角学基于这些关于类似三角形的关键信息。 Suppose we think about all the right triangles in the world with, say, a 41 degree angle.
So here's some random right triangles that have a 41 degree angle. Of course, there are many different size and orientations, but 都有相同的基本形状。这一切的所有41度右三角形都是相似的， 因为它们都共享41度角以及90度角。这是他们共同分享的两个角度。
所以他们必须是相似的，这意味着所有方面都是成比例的。换句话说，我可以找到其中任何一个的比例。 所有这些比率在其余部分中都是相同的。腿部和斜边之间的41度角度。 We will call that leg, the leg that touches the 41 degree angle, the leg that is adjacent to that angle.
另一条腿与41度角相反，所以我们称之为相反的。 So here we have the triangle with the three sides labeled the hypotenuse, the opposite and the adjacent. 现在这里的三个原理比率是正弦比。正弦等，41度的正弦等于斜边的相反。
The cosine equals adjacent over hypotenuse. The tangent equals opposite over adjacent. 学生们经常记住使用助学金索哈卡托的这三个比率。萨赫卡托是什么意思？ Well, SOHCAHTOA, sine is opposite over hypotenuse. That's the SOH.
余弦在斜边邻近;那是cah。切线与邻近相反。 所以，我们必须记住它是SOH，CAH，TOA。请注意，这些中的所有三个都被写为角度41度的函数， because if we changed the angle, all the ratios would be different. Nevertheless, as long as we have a 41 degree right triangle, no matter the size or orientation, all these ratios will be the same.
正弦和余弦和41度的切线，以及任何其他可能的角度已经存储在计算器中。 您只需确保计算器处于学位模式，而不是弧度模式。 We'll talk more about radians in an upcoming video. Therefore, if we're given a right triangle with one known acute angle, and one known length, we can always find the other two lengths.
So, suppose we have this setup. We have a right triangle. We have an angle of ten degrees, a tiny little acute angle. And opposite that ten degree angle, the opposite side, is three centimeters. We want to find the other two lengths, for example. Well, certainly we know that the sin(10°)=opp/hyp=3/AB.
Now if we multiply both sides by AB we get AB*sin(10°) = 3, so we divide by that. 如果我们需要，我们可以在计算器上计算这一点。SIN 10度约为0.1736。 3除以这个数字大约是17.3,这是length of the hypotenuse, AB.
我们也可以找到侧面交流。我们知道10的切线是OPP / adj，这将是3 / AC。 Same thing, multiply by AC/tan(10). Now we can find this in our calculator, tan(10) is about 0.1763, 3 divided by that number is about 17.0. And so we could find the two other lengths, purely from the angle and the one given length.
This is very powerful. Here's a practice problem. 暂停视频，然后我们会谈谈这个。好的，首先要注意到我们这里有三，四，五三角。 这对注意事项非常重要，因为测试通常希望您识别三，四，五三角形。
So that missing side XZ has to equal four. Now notice that we want the tangent of angle X. From the perspective of X, three is the opposite side, and four is the adjacent side. Very important. It would be very different if we were finding the tangent from Y.
But from the point of view from X, three is opposite, and XZ = 4, that's the adjacent. 当然，与相邻相反，所以相反的是YZ，相邻的是XZ，这是3/4。 So it has to be answer choice E. Here's another practice problem.
暂停视频，然后我们会谈谈这个。好的，所以我们被占用了一个角度，我们给了两个长度的SQ和QR，而且 we're also told that the tangent of 35 degrees is approximately 0.700, and we want to know the area of the triangle. 好吧，我们已经知道基础，我们需要高度，我们需要PQ的长度，以弄清楚三角形的区域。
Well, we know that the tangent of 35 degrees, that involves PQ, that's PQ/SQ. Well that's good because we know SQ. That's h/SQ, we need that h. H = 5 x Tan（35度）。在这里，我们可以使用它们给我们的近似，35度的切线为0.7。
Well, 5 x 0.7 is 3.5, so h = 3.5. Very useful. Now that we know h, we can find the area. Of course, the area of a triangle is 1/2bh. So that's 1/2(8), which is the full base from S to R is a length of 8. 1/2(8)(3.5), 1/2 of 8 is 4.
然后4 x 3.5，我们将使用加倍和减半的技巧。1/2的4是2。 3.5为7.5为7. 2 x 7是14。 That's the area. So the area is 14.
In general, for general angel, mathematicians typically use the Greek letter theta. 我们可以使用它来进行一般陈述，真正的任何角度。所以θ的罪是opp / hyp。 余弦是adm / hyp，切线是opp / adj。这是基本的Sohcahtoa模式。
现在，当我们谈论三角形内的角度时，这些都是如此。因此，这意味着θ必须大于0度和 小于90度。它必须在三角形内具有可能的急性值。 Right now, that's where we're gonna focus. In this video, I'll just discuss one more important relationship that you may have 了解测试。
We know of course from the Pythagorean theorem, that the adjacent squared plus the opposite squared, has the equal hypotenuse squared, that's obviously true, because of the Pythagorean theorem. We'll divide each term by hypotenuse squared. On the right side, we'll get a hypotenuse squared divided by hypotenuse squared which is one.
We'll get adjacent squared divided by hypotenuse squared, while adjacent divided by hypotenuse is cosigned. 一个由斜边的相对划分为正弦。所以我们得到余弦方形+正弦方形= 1。 And this is the Pythagorean identity. Notice, incidentally, when we square trig function, we write the square after the name of the function and before the angle.
所以我们把它写成cosθ的平方,或者正弦平方uared theta. So this is an important trig formula, and we'll return to this a few times. This is a very good one to know. Here's another practice problem. 暂停视频并读取此。以下是要选择的表达式。
好好看看这些。并看看你是否可以自己解决问题。 您可以暂停视频，当您准备好时，恢复，我们会一起解决。 好的，让我们想一想。我们将用绳索绘制一个正确的三角形，作为斜边， the horizontal base at the level of the tip of the prow, which is slightly above the water, and the height up to the top of the pole.
Which is at P. Okay. 嗯，从P，Pr的35度角，该段Pr是相邻的一侧。那就是垂直变化帮助我们。 所以我们要需要它。因此，我们需要余弦来汇率邻近斜边。
Is the cosine of 35 degrees is adjacent over hypotenuse; that's PR over 25. So, PR would equal 25 x cos (35 degrees). Very good. So, we have that length; the length of that entire segment, PR. 好吧，PR没有，完全是我们正在寻找的长度。问题特别询问， 高潮和低潮之间的水平变化。
因此，在高潮中，船的船程在D的水平，处于码头表面的水平。 And at low tide, the prow is at the level of R and B, that horizontal line at the bottom of the triangle. So what we need, the change in level, is DR. DR is the difference between high tide and low tide.
Well, we know that PD+ DR = PR. The two little segments together add up to the big segment. 这样，这意味着3 + DR = 25 x COS（35度）。这是我们获得的表达式。 因此，如果我们想要博士，我们从两侧减去三个，这是高度变化的表达。
我们回到答案选择，我们选择了这一点，回答选择C.总之，知道Sohcahtoa是很好的， 这意味着θ的标志与斜边的迹象相反。Theta的余弦是斜边的邻近。 And the tangent is the opposite over the adjacent. For any angle greater than 0 and less than 90 degrees, all the right angles with that acute angle are similar.
And so all these ratios are the same for all of them. So you pick any angle, say 23 degrees, a 23 degree right triangle, 任何23度右三角形，都会与任何其他相似，这就是为什么所有这些比率都是一样的。 And you can find the values for these three ratios on your calculator, although the test often supplies any numbers you need.
Read full transcript
回想一下，如果我们在一个三角形中只知道两个角度等于另一个三角形的两个角度，那么两个三角形必须是类似的 that means that they have the same basic shape. One is just a scaled up, or a scaled down version of the other. 所有三个角度都是相同的，它是相同的基本形状。一旦我们知道这两个三角形是相似的， we know that all their sides are proportional.
It's very easy to show that two triangles are similar, and once we know that, we get a lot of information. 所有三角学基于这些关于类似三角形的关键信息。 Suppose we think about all the right triangles in the world with, say, a 41 degree angle.
So here's some random right triangles that have a 41 degree angle. Of course, there are many different size and orientations, but 都有相同的基本形状。这一切的所有41度右三角形都是相似的， 因为它们都共享41度角以及90度角。这是他们共同分享的两个角度。
所以他们必须是相似的，这意味着所有方面都是成比例的。换句话说，我可以找到其中任何一个的比例。 所有这些比率在其余部分中都是相同的。腿部和斜边之间的41度角度。 We will call that leg, the leg that touches the 41 degree angle, the leg that is adjacent to that angle.
另一条腿与41度角相反，所以我们称之为相反的。 So here we have the triangle with the three sides labeled the hypotenuse, the opposite and the adjacent. 现在这里的三个原理比率是正弦比。正弦等，41度的正弦等于斜边的相反。
The cosine equals adjacent over hypotenuse. The tangent equals opposite over adjacent. 学生们经常记住使用助学金索哈卡托的这三个比率。萨赫卡托是什么意思？ Well, SOHCAHTOA, sine is opposite over hypotenuse. That's the SOH.
余弦在斜边邻近;那是cah。切线与邻近相反。 所以，我们必须记住它是SOH，CAH，TOA。请注意，这些中的所有三个都被写为角度41度的函数， because if we changed the angle, all the ratios would be different. Nevertheless, as long as we have a 41 degree right triangle, no matter the size or orientation, all these ratios will be the same.
正弦和余弦和41度的切线，以及任何其他可能的角度已经存储在计算器中。 您只需确保计算器处于学位模式，而不是弧度模式。 We'll talk more about radians in an upcoming video. Therefore, if we're given a right triangle with one known acute angle, and one known length, we can always find the other two lengths.
So, suppose we have this setup. We have a right triangle. We have an angle of ten degrees, a tiny little acute angle. And opposite that ten degree angle, the opposite side, is three centimeters. We want to find the other two lengths, for example. Well, certainly we know that the sin(10°)=opp/hyp=3/AB.
Now if we multiply both sides by AB we get AB*sin(10°) = 3, so we divide by that. 如果我们需要，我们可以在计算器上计算这一点。SIN 10度约为0.1736。 3除以这个数字大约是17.3,这是length of the hypotenuse, AB.
我们也可以找到侧面交流。我们知道10的切线是OPP / adj，这将是3 / AC。 Same thing, multiply by AC/tan(10). Now we can find this in our calculator, tan(10) is about 0.1763, 3 divided by that number is about 17.0. And so we could find the two other lengths, purely from the angle and the one given length.
This is very powerful. Here's a practice problem. 暂停视频，然后我们会谈谈这个。好的，首先要注意到我们这里有三，四，五三角。 这对注意事项非常重要，因为测试通常希望您识别三，四，五三角形。
So that missing side XZ has to equal four. Now notice that we want the tangent of angle X. From the perspective of X, three is the opposite side, and four is the adjacent side. Very important. It would be very different if we were finding the tangent from Y.
But from the point of view from X, three is opposite, and XZ = 4, that's the adjacent. 当然，与相邻相反，所以相反的是YZ，相邻的是XZ，这是3/4。 So it has to be answer choice E. Here's another practice problem.
暂停视频，然后我们会谈谈这个。好的，所以我们被占用了一个角度，我们给了两个长度的SQ和QR，而且 we're also told that the tangent of 35 degrees is approximately 0.700, and we want to know the area of the triangle. 好吧，我们已经知道基础，我们需要高度，我们需要PQ的长度，以弄清楚三角形的区域。
Well, we know that the tangent of 35 degrees, that involves PQ, that's PQ/SQ. Well that's good because we know SQ. That's h/SQ, we need that h. H = 5 x Tan（35度）。在这里，我们可以使用它们给我们的近似，35度的切线为0.7。
Well, 5 x 0.7 is 3.5, so h = 3.5. Very useful. Now that we know h, we can find the area. Of course, the area of a triangle is 1/2bh. So that's 1/2(8), which is the full base from S to R is a length of 8. 1/2(8)(3.5), 1/2 of 8 is 4.
然后4 x 3.5，我们将使用加倍和减半的技巧。1/2的4是2。 3.5为7.5为7. 2 x 7是14。 That's the area. So the area is 14.
In general, for general angel, mathematicians typically use the Greek letter theta. 我们可以使用它来进行一般陈述，真正的任何角度。所以θ的罪是opp / hyp。 余弦是adm / hyp，切线是opp / adj。这是基本的Sohcahtoa模式。
现在，当我们谈论三角形内的角度时，这些都是如此。因此，这意味着θ必须大于0度和 小于90度。它必须在三角形内具有可能的急性值。 Right now, that's where we're gonna focus. In this video, I'll just discuss one more important relationship that you may have 了解测试。
We know of course from the Pythagorean theorem, that the adjacent squared plus the opposite squared, has the equal hypotenuse squared, that's obviously true, because of the Pythagorean theorem. We'll divide each term by hypotenuse squared. On the right side, we'll get a hypotenuse squared divided by hypotenuse squared which is one.
We'll get adjacent squared divided by hypotenuse squared, while adjacent divided by hypotenuse is cosigned. 一个由斜边的相对划分为正弦。所以我们得到余弦方形+正弦方形= 1。 And this is the Pythagorean identity. Notice, incidentally, when we square trig function, we write the square after the name of the function and before the angle.
所以我们把它写成cosθ的平方,或者正弦平方uared theta. So this is an important trig formula, and we'll return to this a few times. This is a very good one to know. Here's another practice problem. 暂停视频并读取此。以下是要选择的表达式。
好好看看这些。并看看你是否可以自己解决问题。 您可以暂停视频，当您准备好时，恢复，我们会一起解决。 好的，让我们想一想。我们将用绳索绘制一个正确的三角形，作为斜边， the horizontal base at the level of the tip of the prow, which is slightly above the water, and the height up to the top of the pole.
Which is at P. Okay. 嗯，从P，Pr的35度角，该段Pr是相邻的一侧。那就是垂直变化帮助我们。 所以我们要需要它。因此，我们需要余弦来汇率邻近斜边。
Is the cosine of 35 degrees is adjacent over hypotenuse; that's PR over 25. So, PR would equal 25 x cos (35 degrees). Very good. So, we have that length; the length of that entire segment, PR. 好吧，PR没有，完全是我们正在寻找的长度。问题特别询问， 高潮和低潮之间的水平变化。
因此，在高潮中，船的船程在D的水平，处于码头表面的水平。 And at low tide, the prow is at the level of R and B, that horizontal line at the bottom of the triangle. So what we need, the change in level, is DR. DR is the difference between high tide and low tide.
Well, we know that PD+ DR = PR. The two little segments together add up to the big segment. 这样，这意味着3 + DR = 25 x COS（35度）。这是我们获得的表达式。 因此，如果我们想要博士，我们从两侧减去三个，这是高度变化的表达。
我们回到答案选择，我们选择了这一点，回答选择C.总之，知道Sohcahtoa是很好的， 这意味着θ的标志与斜边的迹象相反。Theta的余弦是斜边的邻近。 And the tangent is the opposite over the adjacent. For any angle greater than 0 and less than 90 degrees, all the right angles with that acute angle are similar.
And so all these ratios are the same for all of them. So you pick any angle, say 23 degrees, a 23 degree right triangle, 任何23度右三角形，都会与任何其他相似，这就是为什么所有这些比率都是一样的。 And you can find the values for these three ratios on your calculator, although the test often supplies any numbers you need.
