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Intro to SOHCAHTOA
成绩单
三角学习, Introduction to SOHCAHTOA. This lesson assumes that you are familiar with the ideas of similar triangles,
覆盖在几何模块中。如果类似三角形的想法绝对不熟悉你，
在观看三角仪视频之前，在几何模块中观看该视频可能会有所帮助。
Recall that if we know just two angles in one triangle are equal to two angles in the other triangle, then the two triangles must be similar and 这意味着它们具有相同的基本形状。一个只是一个缩放的，或另一个缩小版本。 All three angles are the same, and it's the same basic shape. Once we know that the two triangles are similar, 我们知道他们的所有方面都是成比例的。
很容易表明两个三角形是相似的，一旦我们知道，我们就会得到很多信息。 All of trigonometry is based on these key pieces of information about similar triangles. 假设我们思考世界上所有正确的三角形，比如为41度角。
所以这是一些随机右三角形，具有41度角。当然，有许多不同的大小和方向，但 all have the same basic shape. All of these 41 degree right triangles are similar, because they all share a 41 degree angle as well as a 90 degree angle. That's two angles they share in common.
So they have to be similar, and this means all the sides are proportional. In other words, I could find the ratio in any one of them. And all these same ratios would be the same in all the rest of them. The 41 degree angle's between a leg and a hypotenuse. 我们会称之为腿，触摸41度角的腿，腿部与该角度相邻。
The other leg is opposite from the 41 degree angle, so we call that the opposite. 所以在这里，我们有三角形的三面标记为斜边，相反和相邻的。 Now the three principle ratios here are the sine ratio. Sine equal, sine of 41 degrees equals opposite over hypotenuse.
余弦等于斜边等。切线等于相邻的相反。 Students often remember those three ratios using the mnemonic SOHCAHTOA. What is meant by SOHCAHTOA? 嗯，Sohcahtoa，正弦在斜边。那是豪君。
Cosine is adjacent over hypotenuse; that's the CAH. And tangent is the opposite over adjacent. So, we have to remember that it's SOH, CAH, TOA. Notice that all three of these are written as functions of the angle 41 degrees, 因为如果我们改变了角度，所以所有的比率都会不同。尽管如此，只要我们有一个41度的右三角形， 无论尺寸还是定向，所有这些比率都会相同。
The sine and cosine and tangent of 41 degrees, and of any other possible angle are already stored in your calculator. You just have to make sure that your calculator is in degrees mode, instead of radians mode. 我们将更多地讨论即将到来的视频中的弧度。因此，如果我们被赋予一种具有一个已知锐角的正确三角形，并且 一个已知的长度，我们总能找到另外两个长度。
所以，假设我们有这个设置。我们有一个正确的三角形。 我们的角度为10度，一点小锐角。而且相反十个度角，相反的一侧是三厘米。 例如，我们想找到另外两个长度。好吧，当然我们知道 SIN（10°）= OPP / hyp = 3 / ab。
现在，如果我们将双方乘以AB，我们得到AB * SIN（10°）= 3，所以我们划分。 And if we needed, we can compute this on a calculator. Sin 10 degrees is about 0.1736. 3除以该数字约为17.3，这是斜边，ab的长度。
We could also find side AC. We know that the tangent of 10 is opp/adj, this would be 3/AC. 同样的事情，乘以AC / TAN（10）。现在我们可以在我们的计算器中找到它，Tan（10）是约0.1763， 3 divided by that number is about 17.0. And so we could find the two other lengths, 纯粹从角度和一个给定的长度。
这是非常强大的。这是一个练习问题。 Pause the video, and then we'll talk about this. Okay, the first thing to notice is what we have here is a three, four, five triangle. That's very important to notice because the test will often expect you to recognize a three, four, five triangle.
因此，缺少侧面XZ必须等于四个。现在注意到我们想要角度X的切线。 从X的角度来看，三个是相对的一侧，四个是相邻的一侧。 很重要。如果我们找到Y的切线，那将是非常不同的。
But from the point of view from X, three is opposite, and XZ = 4, that's the adjacent. 当然切对相邻,所以the opposite is YZ, the adjacent is XZ, and that is 3/4. 所以它必须是回答选择E.这是另一个练习问题。
Pause the video and then we'll talk about this. Okay, so we're given an angle, and we're given two lengths SQ and QR, and 我们也被告知，35度的切线约为0.700，我们希望了解三角形的区域。 Well we already know the base, we need the height, we need the length of PQ, in order to figure out the area of the triangle.
Well, we know that the tangent of 35 degrees, that involves PQ, that's PQ/SQ. 好吧，这很好，因为我们知道SQ。那是h / sq，我们需要h。 h = 5 x tan(35 degrees). And here we can use the approximation they give us, tangent of 35 degrees is 0.7.
嗯，5 x 0.7是3.5，所以h = 3.5。很有用。 现在我们知道H，我们可以找到该地区。当然，三角形的面积为1 / 2bh。 因此，1/2（8），这是S到R的完全基座的长度为8. 1/2（8）（3.5），1/2的8是4。
Then for 4 x 3.5, we'll use the doubling and halving trick. 1/2 of 4 is 2. Double the 3.5 is 7. 2 x 7 is 14. 那是该地区。所以该地区是14岁。
一般来说，对于一般天使，数学家通常使用希腊字母θ。 We can use this to make general statements, true for any angle. So the sin of theta is opp/hyp. The cosine is adj/hyp, and the tangent is opp/adj. And this is the basic SOHCAHTOA pattern.
Right now, these are true when we are talking about angles inside triangles. So that means theta would have to be greater than 0 degrees and less than 90 degrees. It would have to have a possible acute value inside a triangle. 现在，这就是我们要关注的地方。在这个视频中，我会讨论你可能拥有的一个更重要的关系 to know on the test.
我们知道从毕达哥兰定理，那个相邻的平方加上相反的平方，具有平等的斜边，这显然是真的， 因为毕达哥拉斯定理。我们将通过斜边的平方分开。 在右侧，我们将获得一个斜边的平方，除以斜边的平方。
我们将获得相邻的平方除以斜边的平方，而近距离除以斜边的副本。 An opposite divided by a hypotenuse is sine. So we get cosine squared + sine squared = 1. 这是毕达哥拉斯的身份。通信情况下，当我们方形触发功能时， 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. 这是一个很好的知识。这是另一个练习问题。 Pause the video and read this. Here are the expressions from which to choose.
Take a good look at these. And see if you can solve the problem on your own. You can pause the video and when you're ready, resume, and we'll solve it together. Okay, let's think about this. We're gonna draw a right triangle with the rope as the hypotenuse, 尺寸尖端水平的水平底座，其略高于水面，以及杆顶部的高度。
这是在P.好的。 Well, from the 35 degree angle at P, PR, that segment PR is the adjacent side. And that's gonna help us with the vertical change. So we're gonna need that. So we need the cosine to relate the adjacent by hypotenuse.
是35度的余弦，邻近斜边;这是超过25的PR。所以，PR将等于25 x cos（35度）。 非常好。所以，我们的长度;整个段，公关的长度。 Well, PR does not, exactly the length that we're looking for. The question asks very specifically, the change in level between high tide and low tide.
So at high tide, the prow of the boat was at the level of D, was at the level of the surface of the dock. 并且在低潮中，船舶处于R和B的水平，三角形底部的水平线。 所以我们需要的是，水平的变化是博士。DR是高潮和低潮之间的区别。
好吧，我们知道PD + DR = PR。这两个小部分在一起加起来大部分。 So that means that 3 + DR = 25 x cos(35 degrees). That's the expression we got for PR. So if we want DR, we subtract three from both sides, and that's the expression for the change in height.
And we go back to the answer choices, and we choose this one, answer choice C. In summary, it's good to know SOHCAHTOA, which means that the sign of theta is the opposite over hypotenuse. The cosine of theta is the adjacent over the hypotenuse. And the tangent is the opposite over the adjacent. For any angle greater than 0 and 小于90度，与锐角的所有直角都是相似的。
所以所有这些比率都是相同的。所以你选择任何角度，说23度，23度右三角形， any 23 degree right triangle, is going to be similar to any other, and that's why all of these ratios are the same. And you can find the values for these three ratios on your calculator, although the test often supplies any numbers you need.
阅读完整成绩单
Recall that if we know just two angles in one triangle are equal to two angles in the other triangle, then the two triangles must be similar and 这意味着它们具有相同的基本形状。一个只是一个缩放的，或另一个缩小版本。 All three angles are the same, and it's the same basic shape. Once we know that the two triangles are similar, 我们知道他们的所有方面都是成比例的。
很容易表明两个三角形是相似的，一旦我们知道，我们就会得到很多信息。 All of trigonometry is based on these key pieces of information about similar triangles. 假设我们思考世界上所有正确的三角形，比如为41度角。
所以这是一些随机右三角形，具有41度角。当然，有许多不同的大小和方向，但 all have the same basic shape. All of these 41 degree right triangles are similar, because they all share a 41 degree angle as well as a 90 degree angle. That's two angles they share in common.
So they have to be similar, and this means all the sides are proportional. In other words, I could find the ratio in any one of them. And all these same ratios would be the same in all the rest of them. The 41 degree angle's between a leg and a hypotenuse. 我们会称之为腿，触摸41度角的腿，腿部与该角度相邻。
The other leg is opposite from the 41 degree angle, so we call that the opposite. 所以在这里，我们有三角形的三面标记为斜边，相反和相邻的。 Now the three principle ratios here are the sine ratio. Sine equal, sine of 41 degrees equals opposite over hypotenuse.
余弦等于斜边等。切线等于相邻的相反。 Students often remember those three ratios using the mnemonic SOHCAHTOA. What is meant by SOHCAHTOA? 嗯，Sohcahtoa，正弦在斜边。那是豪君。
Cosine is adjacent over hypotenuse; that's the CAH. And tangent is the opposite over adjacent. So, we have to remember that it's SOH, CAH, TOA. Notice that all three of these are written as functions of the angle 41 degrees, 因为如果我们改变了角度，所以所有的比率都会不同。尽管如此，只要我们有一个41度的右三角形， 无论尺寸还是定向，所有这些比率都会相同。
The sine and cosine and tangent of 41 degrees, and of any other possible angle are already stored in your calculator. You just have to make sure that your calculator is in degrees mode, instead of radians mode. 我们将更多地讨论即将到来的视频中的弧度。因此，如果我们被赋予一种具有一个已知锐角的正确三角形，并且 一个已知的长度，我们总能找到另外两个长度。
所以，假设我们有这个设置。我们有一个正确的三角形。 我们的角度为10度，一点小锐角。而且相反十个度角，相反的一侧是三厘米。 例如，我们想找到另外两个长度。好吧，当然我们知道 SIN（10°）= OPP / hyp = 3 / ab。
现在，如果我们将双方乘以AB，我们得到AB * SIN（10°）= 3，所以我们划分。 And if we needed, we can compute this on a calculator. Sin 10 degrees is about 0.1736. 3除以该数字约为17.3，这是斜边，ab的长度。
We could also find side AC. We know that the tangent of 10 is opp/adj, this would be 3/AC. 同样的事情，乘以AC / TAN（10）。现在我们可以在我们的计算器中找到它，Tan（10）是约0.1763， 3 divided by that number is about 17.0. And so we could find the two other lengths, 纯粹从角度和一个给定的长度。
这是非常强大的。这是一个练习问题。 Pause the video, and then we'll talk about this. Okay, the first thing to notice is what we have here is a three, four, five triangle. That's very important to notice because the test will often expect you to recognize a three, four, five triangle.
因此，缺少侧面XZ必须等于四个。现在注意到我们想要角度X的切线。 从X的角度来看，三个是相对的一侧，四个是相邻的一侧。 很重要。如果我们找到Y的切线，那将是非常不同的。
But from the point of view from X, three is opposite, and XZ = 4, that's the adjacent. 当然切对相邻,所以the opposite is YZ, the adjacent is XZ, and that is 3/4. 所以它必须是回答选择E.这是另一个练习问题。
Pause the video and then we'll talk about this. Okay, so we're given an angle, and we're given two lengths SQ and QR, and 我们也被告知，35度的切线约为0.700，我们希望了解三角形的区域。 Well we already know the base, we need the height, we need the length of PQ, in order to figure out the area of the triangle.
Well, we know that the tangent of 35 degrees, that involves PQ, that's PQ/SQ. 好吧，这很好，因为我们知道SQ。那是h / sq，我们需要h。 h = 5 x tan(35 degrees). And here we can use the approximation they give us, tangent of 35 degrees is 0.7.
嗯，5 x 0.7是3.5，所以h = 3.5。很有用。 现在我们知道H，我们可以找到该地区。当然，三角形的面积为1 / 2bh。 因此，1/2（8），这是S到R的完全基座的长度为8. 1/2（8）（3.5），1/2的8是4。
Then for 4 x 3.5, we'll use the doubling and halving trick. 1/2 of 4 is 2. Double the 3.5 is 7. 2 x 7 is 14. 那是该地区。所以该地区是14岁。
一般来说，对于一般天使，数学家通常使用希腊字母θ。 We can use this to make general statements, true for any angle. So the sin of theta is opp/hyp. The cosine is adj/hyp, and the tangent is opp/adj. And this is the basic SOHCAHTOA pattern.
Right now, these are true when we are talking about angles inside triangles. So that means theta would have to be greater than 0 degrees and less than 90 degrees. It would have to have a possible acute value inside a triangle. 现在，这就是我们要关注的地方。在这个视频中，我会讨论你可能拥有的一个更重要的关系 to know on the test.
我们知道从毕达哥兰定理，那个相邻的平方加上相反的平方，具有平等的斜边，这显然是真的， 因为毕达哥拉斯定理。我们将通过斜边的平方分开。 在右侧，我们将获得一个斜边的平方，除以斜边的平方。
我们将获得相邻的平方除以斜边的平方，而近距离除以斜边的副本。 An opposite divided by a hypotenuse is sine. So we get cosine squared + sine squared = 1. 这是毕达哥拉斯的身份。通信情况下，当我们方形触发功能时， 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. 这是一个很好的知识。这是另一个练习问题。 Pause the video and read this. Here are the expressions from which to choose.
Take a good look at these. And see if you can solve the problem on your own. You can pause the video and when you're ready, resume, and we'll solve it together. Okay, let's think about this. We're gonna draw a right triangle with the rope as the hypotenuse, 尺寸尖端水平的水平底座，其略高于水面，以及杆顶部的高度。
这是在P.好的。 Well, from the 35 degree angle at P, PR, that segment PR is the adjacent side. And that's gonna help us with the vertical change. So we're gonna need that. So we need the cosine to relate the adjacent by hypotenuse.
是35度的余弦，邻近斜边;这是超过25的PR。所以，PR将等于25 x cos（35度）。 非常好。所以，我们的长度;整个段，公关的长度。 Well, PR does not, exactly the length that we're looking for. The question asks very specifically, the change in level between high tide and low tide.
So at high tide, the prow of the boat was at the level of D, was at the level of the surface of the dock. 并且在低潮中，船舶处于R和B的水平，三角形底部的水平线。 所以我们需要的是，水平的变化是博士。DR是高潮和低潮之间的区别。
好吧，我们知道PD + DR = PR。这两个小部分在一起加起来大部分。 So that means that 3 + DR = 25 x cos(35 degrees). That's the expression we got for PR. So if we want DR, we subtract three from both sides, and that's the expression for the change in height.
And we go back to the answer choices, and we choose this one, answer choice C. In summary, it's good to know SOHCAHTOA, which means that the sign of theta is the opposite over hypotenuse. The cosine of theta is the adjacent over the hypotenuse. And the tangent is the opposite over the adjacent. For any angle greater than 0 and 小于90度，与锐角的所有直角都是相似的。
所以所有这些比率都是相同的。所以你选择任何角度，说23度，23度右三角形， any 23 degree right triangle, is going to be similar to any other, and that's why all of these ratios are the same. And you can find the values for these three ratios on your calculator, although the test often supplies any numbers you need.
