三角学。Sohcahtoa介绍。 本课程假设您熟悉几何模块中覆盖的类似三角形的思想。 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.

Recall that if we know that just two angles in one triangle are equal to two angles in the other triangle, then the two triangle must be similar. 这意味着它们具有相同的基本形状，一个只是一个缩放或缩放版本的另一个。 所有三个角度都是一样的。它是相同的基本形状。

Once we know that the two triangles are similar, 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 are some random right triangles that have a 41 degree angle. Of course, there are many different sizes and orientations, but all have the same basic shape. All of these 41 degree right triangles are similar, 因为它们都共享41度角，以及90度角。

That's two angles they share in common, so they have to be similar. This means all the sides are proportional. 换句话说，我可以找到其中任何一个的比例，并且所有这些比率在它们的所有其他比率都是相同的。 41度角是一条腿和一个hypoten之间use. 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。 所以在这里，我们的三角形有三角形标记，斜边相反，邻近。 Now, the three principle ratios here are the sine ratio, sine equals, sine of 41 degrees equals opposite over hypotenuse.

The cosine equals adjacent over hypotenuse. The tangent equals opposite over adjacent. Students often remember those three ratios using the mnemonic SOHCAHTOA. What is meant by SOHCAHTOA? 嗯，Sohcahtoa，正弦在斜边，这是S-O-H。余弦在斜边旁边，这是C-A-H。

And tangent is the opposite over adjacent, so we have to remember that's it's SOH CAH TOA. 请注意，所有这些都作为角度，41度的函数写入，因为如果我们改变了角度，则所有比率将不同。 尽管如此，只要我们拥有41度右三角形，无论大小或方向，所有这些比率都会相同。

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。 You just have to make sure that your calculator is in degrees mode instead of radians mode. We'll talk more about radians in an upcoming video. Therefore, if we are 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 10 degrees, a tiny little acute angle, and 相反的10度角，相对的一侧是3厘米。例如，我们希望找到另外两个长度。 当然，我们知道10度的罪是斜边的对面。所以这将是3副作用。

现在，如果我们通过AB乘以两侧，我们将AB次如Sine 10等于3除以正弦10。 Sine 10 is sum number. So we divide by that. And if we needed we could compute this on a calculator. Sine 10 degrees is about 0.1736.

3 divided by that number is about 17.3. That's the length of the hypotenuse AB. We could also find side AC. We know that the tangent of 10 is opposite over adjacent. 这将是3的ac。同样的事情，乘法乘以晒黑（10）除以棕褐色（10）。

现在，我们可以在我们的计算器上找到它，Tan（10）约为0.1763。3除以该数字约为17.0。 所以，我们可以纯粹从角度找到另外两个长度，而这是一个给定的长度，这是非常强大的。 Here's a practice problem. Pause the video, and then we'll talk about this.

首先要注意到我们这里有3,4,5三角形。这对注意非常重要， because the test will often expect you to recognize a 3, 4, 5 triangle. So that missing side XZ has to equal 4. Now, notice that we want the tangent of angle x. From the perspective of x, 3 is the opposite side, and 4 is the adjacent side.

Very important. It would be very different if we were finding the tangent from y. 但从X的角度来看，3的角度为相反，XZ等于4.相邻。 And, of course, tangent is opposite over adjacent, so the opposite is YZ, the adjacent is XZ, and that is 3 over 4.

所以它必须是回答选择E. [空白音频]这是另一个练习问题。 Pause the video, and then we'll talk about this. So we're given an angle, we're given two lengths SQ and QR, and we're also told that the tangent of 35 degrees is approximately 0.700, and we want to know the area of the triangle.

Well, we already know the base. We need the height. 我们需要PQ的长度，以找出三角形的区域。嗯，我们知道35度的切线，涉及PQ。 这是平方的pq。好吧，这很好，因为我们知道SQ。

That's h over SQ, and we need that h. H equals 5 times tangent of 35 degrees, and 在这里，我们可以给我们的近似，35度的切线为0.7。嗯，5次（0.7）是3.5，所以H = 3.5。 Very useful, now that we know h. We can find the area.

Of course, the area of a triangle is one-half base times height. So that's one-half (8), which is the full base from S to R is the length of 8. One-half (8) times 3.5. One-half of 8 is 4. 然后，对于4 x 3.5，我们将使用加倍和减半的触发。一半是2。

3.5的双倍为7. 2次7是14。 That's the area. So the area is 14. In general, for a general angle, mathematicians typically use the Greek letter theta.

We can use this to make general statements, true for any angle. So the sine of theta is opposite of our hypotenuse. 余弦在斜边旁边相邻，并且切线与相邻相反。这是基本的Sohcahtoa模式。 Right now, these are true when we're talking about angles inside triangles. So that means theta would have to be greater than 0 degrees and 小于90度。

It would have to have a possible acute value inside a triangle. Right now, that's where we're gonna focus. In this video, I'll just discuss one more important relationship that you may have to know on the test. We know, of course, from the Pythagorean theorem that the adjacent squared plus the opposite squared has to equal the hypotenuse squared.

That's obviously true, because of the Pythagorean theorem. We'll divide each term by hypotenuse squared. 在右侧，我们将获得一个斜边的平方，它是一个斜边的平方分为1。 我们将获得相邻的平方除以斜边的平方。嗯，邻近的斜边是余弦，和 对面除以斜边是正弦。

所以我们得到余弦方形加正弦方形等于1.这是毕达哥拉斯的身份。 Notice, incidentally, when we square a trig function, we write the square after the name of the function and before the angle, so 我们将其写成余弦平方θ或正弦方形θ。所以这是一个重要的三角形公式，我们将返回这一点。

但这是一个很好的知识。这是另一个练习问题。 所以暂停视频并阅读这一点。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，恢复并我们将在一起解决。让我们想一想。 We're gonna draw a right triangle with the rope as the hypotenuse, the horizontal base at the level of the tip of the prow which is slightly 出水面,和高度的the pole which is that P. Well, from the 35 degree angle at P, PR, that segment PR is the adjacent side.

那就是垂直变化帮助我们，所以我们需要这一点。因此，余弦，我们需要余弦与斜边的邻近。 35度的余弦在斜边旁边。该PR超过25，所以PR将等于余弦的25倍，为35度。 非常好，所以我们的长度，整个段的长度，公关。好吧，PR，这不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 plus DR equals the length of PR, the 2 little segments together add up to the big segment, so that means that 3 plus DR equals 25 times cosine of 35 degrees. That's the expression we got for PR.

So if we want DR we subtract 3 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. [BLANK AUDIO] In summary, it's good to know SOHCAHTOA, which means that the sign of theta is the opposite over hypotenuse.

Theta的余弦是斜边的邻近。并且切线与相邻的相对。 对于任何大于0和小于90度的角度，具有锐角的所有直角都是相似的。 And so all these ratios are the same for all of them. So if you pick any angle, say 23 degrees.

一个23度右三角形，任何23度右三角形的就会与任何其他相似。 And that's why all of these ratios are the same. And you can find the values for these three ratios on your calculator, 虽然测试通常提供您需要的任何数字。