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Advanced Trig Formulas
Transcript
Now we can talk about some advanced trig formulas. In this video I will discuss a few advanced formulas from trigonometry.
作为一般规则，您不需要在此视频中记住任何内容。我要告诉你一些复杂的公式。
You simply need to be able to work with the formulas if and when the test presents you with it.
Having some familiarity with these formulas ahead of time will make that much easier. First of all, what I'll call the sign rules. This first category has the rules for when the positive and the negative signs of x changes. The sine(-x) = -sine(x), but the cosine(-x) = just cosine(x).
In the unit circle starting from zero if we move some angle clockwise, and then the same angle counterclockwise, we will arrive at points that have 对面的Y坐标和相同的X坐标。所以换句话说，我们将以这种方式举动。 并且那两个点具有完全相同的X坐标，但它们具有与Y坐标相对的。
And so that's why the sines, the sine(x) are negatives of each other, but the cosines(x) are identical. These formulas have implications for the shape of the graph. The standard cosine graph is a reflection of itself over the y-axis. The sine graph is an image of itself under 180 degrees rotational symmetry around the origin.
Now if you're familiar with the ideas of an even function and an odd function, cosine is an even function and sine is an odd function. 这是一个convenient thing to know if you understand that, but the ACT does not ask about those kinds of symmetries. 接下来，我们将讨论角度加法和减法规则。我们可以为三个急性角度推导出正弦和余弦的确切值， pi over 6, pi over 4, and pi over 3.
Those are our angles in the special triangles. If we add and subtract these angles in various combinations, we can get a few more angles. And mathematicians have derived the formulas for the sine or cosine of the sum or difference of two known angles. Assume that alpha and beta are angles for which we know the values of sine and cosine.
这四种配方是正弦（α+β），正弦（α-β），余弦（α+β）和余弦（α-β）。 So, again, you do not have to have these four complicated formulas memorized. The test will give you one of these if you're expected to know it. 但是，与他们一起练习是有用的，所以如果你被给出一个问题，它很熟悉。
这是一个练习问题。暂停视频，然后我们会谈谈这个。 Okay, for QI angles, quadrant I angles alpha and beta, the sine(alpha) is three-fifths and the cosine(beta) is twelve-thirteenths. Given that, find the cosine(alpha + beta).
Well, the first thing we need to do, is we need to recognize that we're dealing with some very important Pythagorean triplets. 如果毕达哥拉斯三胞胎的想法不熟悉，我建议返回并观察视频右三角形，在几何部分。 We're dealing with these Pythagorean Triplets, 3, 4, 5 and 5, 12, 13. So notice that alpha has a sine of three-fifths, so we see that the adjacent leg is 4.
Beta has a cosine of twelve-thirteenths, so the opposite leg is 5. And this means that we can find the cosine(alpha), that's four-fifths and the sine(beta), that's five-thirteenths. So now that we have these four values, we can plug into the formula. They give us the formula cosine(alpha + beta) = cosine alpha cosine beta- sine alpha sine beta.
So we plug all these values in, multiply. We get forty-eight sixty-fifths- fifteen sixty-fifths, which is thirty-three sixty-fifths. And answer choice A is the answer. Next, we'll talk about formulas for non-right triangles. The SOHCAHTOA relationships are wonderful for solving the sides of right triangles, but most triangles in geometry and many triangles in real life are not right triangles.
为此，我们将遵循顶点由大写字母表示的约定，该大写字母也用作角度名。 And each side is the lower case of the same letter as its opposite vertex. So for example, here we see we have the three vertices, A B and C, and opposite from the angle is the side indicated by the lowercase letter of the same letter. So for any triangle ABC, there are two important rules for these non-right triangles.
One of them is the Law of Cosines, which is kind of a generalized version of the Pythagorean theorem, and then the Law of Sines. 所以给出了组合SAS中的数值，也就是说侧面，侧和夹角，或角度，角度和附带的侧。 Or AAS, two angles and a non-included side, or just all three sides, SSS. Now, notice those are the four combinations that determine a triangle.
They're good enough for triangle congruence. So if we're given numerical values in any of those combinations, we could find all the other angles and sides of the triangle. Here's a practice problem. 暂停视频，然后我们会谈谈这个。好的，所以在这里，我们在这里，一边，一边。
我们给了三个边长。我们希望找到角度C的余弦，余弦的较大角度。 从图中出现的那种角度是钝角，角度大于90度。 So we're actually expecting that the cosine of it will be negative. So that's just a prediction.
Let's see how this is borne out by the numbers. Plugging in, we get 2 squared, which is 4 + 3 squared, which is 9- 12 cosine C = 16. Subtract the 9 and the 4. 所以我们得到-12余弦c = 16-9-4，这是3.除以-12，我们得到余弦（c）= 3除以-12， or -3 over 12, which is- one-quarter.
So indeed, the cosine is negative. And the answer is D. You do not need to have the rules discussed here memorized. But it's good to do enough practice problems with them, so 您熟悉它们，使用它们很舒服。这样如果你有问题，那么测试把你的公式和 says use this, you'll already be comfortable with it.
Read full transcript
Having some familiarity with these formulas ahead of time will make that much easier. First of all, what I'll call the sign rules. This first category has the rules for when the positive and the negative signs of x changes. The sine(-x) = -sine(x), but the cosine(-x) = just cosine(x).
In the unit circle starting from zero if we move some angle clockwise, and then the same angle counterclockwise, we will arrive at points that have 对面的Y坐标和相同的X坐标。所以换句话说，我们将以这种方式举动。 并且那两个点具有完全相同的X坐标，但它们具有与Y坐标相对的。
And so that's why the sines, the sine(x) are negatives of each other, but the cosines(x) are identical. These formulas have implications for the shape of the graph. The standard cosine graph is a reflection of itself over the y-axis. The sine graph is an image of itself under 180 degrees rotational symmetry around the origin.
Now if you're familiar with the ideas of an even function and an odd function, cosine is an even function and sine is an odd function. 这是一个convenient thing to know if you understand that, but the ACT does not ask about those kinds of symmetries. 接下来，我们将讨论角度加法和减法规则。我们可以为三个急性角度推导出正弦和余弦的确切值， pi over 6, pi over 4, and pi over 3.
Those are our angles in the special triangles. If we add and subtract these angles in various combinations, we can get a few more angles. And mathematicians have derived the formulas for the sine or cosine of the sum or difference of two known angles. Assume that alpha and beta are angles for which we know the values of sine and cosine.
这四种配方是正弦（α+β），正弦（α-β），余弦（α+β）和余弦（α-β）。 So, again, you do not have to have these four complicated formulas memorized. The test will give you one of these if you're expected to know it. 但是，与他们一起练习是有用的，所以如果你被给出一个问题，它很熟悉。
这是一个练习问题。暂停视频，然后我们会谈谈这个。 Okay, for QI angles, quadrant I angles alpha and beta, the sine(alpha) is three-fifths and the cosine(beta) is twelve-thirteenths. Given that, find the cosine(alpha + beta).
Well, the first thing we need to do, is we need to recognize that we're dealing with some very important Pythagorean triplets. 如果毕达哥拉斯三胞胎的想法不熟悉，我建议返回并观察视频右三角形，在几何部分。 We're dealing with these Pythagorean Triplets, 3, 4, 5 and 5, 12, 13. So notice that alpha has a sine of three-fifths, so we see that the adjacent leg is 4.
Beta has a cosine of twelve-thirteenths, so the opposite leg is 5. And this means that we can find the cosine(alpha), that's four-fifths and the sine(beta), that's five-thirteenths. So now that we have these four values, we can plug into the formula. They give us the formula cosine(alpha + beta) = cosine alpha cosine beta- sine alpha sine beta.
So we plug all these values in, multiply. We get forty-eight sixty-fifths- fifteen sixty-fifths, which is thirty-three sixty-fifths. And answer choice A is the answer. Next, we'll talk about formulas for non-right triangles. The SOHCAHTOA relationships are wonderful for solving the sides of right triangles, but most triangles in geometry and many triangles in real life are not right triangles.
为此，我们将遵循顶点由大写字母表示的约定，该大写字母也用作角度名。 And each side is the lower case of the same letter as its opposite vertex. So for example, here we see we have the three vertices, A B and C, and opposite from the angle is the side indicated by the lowercase letter of the same letter. So for any triangle ABC, there are two important rules for these non-right triangles.
One of them is the Law of Cosines, which is kind of a generalized version of the Pythagorean theorem, and then the Law of Sines. 所以给出了组合SAS中的数值，也就是说侧面，侧和夹角，或角度，角度和附带的侧。 Or AAS, two angles and a non-included side, or just all three sides, SSS. Now, notice those are the four combinations that determine a triangle.
They're good enough for triangle congruence. So if we're given numerical values in any of those combinations, we could find all the other angles and sides of the triangle. Here's a practice problem. 暂停视频，然后我们会谈谈这个。好的，所以在这里，我们在这里，一边，一边。
我们给了三个边长。我们希望找到角度C的余弦，余弦的较大角度。 从图中出现的那种角度是钝角，角度大于90度。 So we're actually expecting that the cosine of it will be negative. So that's just a prediction.
Let's see how this is borne out by the numbers. Plugging in, we get 2 squared, which is 4 + 3 squared, which is 9- 12 cosine C = 16. Subtract the 9 and the 4. 所以我们得到-12余弦c = 16-9-4，这是3.除以-12，我们得到余弦（c）= 3除以-12， or -3 over 12, which is- one-quarter.
So indeed, the cosine is negative. And the answer is D. You do not need to have the rules discussed here memorized. But it's good to do enough practice problems with them, so 您熟悉它们，使用它们很舒服。这样如果你有问题，那么测试把你的公式和 says use this, you'll already be comfortable with it.
