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Advanced Trig Formulas
Now we can talk about some advanced trig formulas. In this video I will discuss a few advanced formulas from trigonometry. As a general rule, you don't need to have anything in this video memorized. I'm gonna show you some complicated formulas. 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 opposite y-coordinates and the same x-coordinates. So in other words, we're gonna move this way and this way. And those two points have the exact same x-coordinate, but they have opposite y-coordinates.
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. That's a convenient thing to know if you understand that, but the ACT does not ask about those kinds of symmetries. Next we'll talk about the angle addition and subtraction rules. We can derive the exact values of sine and cosine for three acute angles, 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.
These four formulas are the sine(alpha + beta), the sine(alpha- beta), the cosine(alpha + beta), and the cosine(alpha- beta). 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. But it's useful to practice with them so if you are given one in a problem, it is familiar.
Here's a practice problem. Pause the video and then we'll talk about this. 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. And if the idea of Pythagorean triplets are not familiar to you, I'd suggest go back and watch the video Right Triangle, in the section on geometry. 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.
For this, we'll follow the conventions that vertices are denoted by capital letters, which also serve as the angle names. 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. So given the numerical values in a combination SAS, that is to say side, side, and an included angle, or ASA, angle, angle and an included side. 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. Pause the video and then we'll talk about this. Okay, so here we're given side, side, side.
We're given the three side lengths. And we want to find the cosine of angle C, cosine of that larger angle. It kind of appears from the diagram that that angle is an obtuse angle, an angle greater than 90 degrees. 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. So we get -12 cosine C = 16- 9- 4, which is 3. Divide by -12, we get cosine(C) = 3 divided by -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 you are familiar with them, and are comfortable with using them. So that way if you have a problem, the test hands you the formula and says use this, you'll already be comfortable with it.
Read full transcriptHaving 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 opposite y-coordinates and the same x-coordinates. So in other words, we're gonna move this way and this way. And those two points have the exact same x-coordinate, but they have opposite y-coordinates.
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. That's a convenient thing to know if you understand that, but the ACT does not ask about those kinds of symmetries. Next we'll talk about the angle addition and subtraction rules. We can derive the exact values of sine and cosine for three acute angles, 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.
These four formulas are the sine(alpha + beta), the sine(alpha- beta), the cosine(alpha + beta), and the cosine(alpha- beta). 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. But it's useful to practice with them so if you are given one in a problem, it is familiar.
Here's a practice problem. Pause the video and then we'll talk about this. 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. And if the idea of Pythagorean triplets are not familiar to you, I'd suggest go back and watch the video Right Triangle, in the section on geometry. 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.
For this, we'll follow the conventions that vertices are denoted by capital letters, which also serve as the angle names. 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. So given the numerical values in a combination SAS, that is to say side, side, and an included angle, or ASA, angle, angle and an included side. 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. Pause the video and then we'll talk about this. Okay, so here we're given side, side, side.
We're given the three side lengths. And we want to find the cosine of angle C, cosine of that larger angle. It kind of appears from the diagram that that angle is an obtuse angle, an angle greater than 90 degrees. 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. So we get -12 cosine C = 16- 9- 4, which is 3. Divide by -12, we get cosine(C) = 3 divided by -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 you are familiar with them, and are comfortable with using them. So that way if you have a problem, the test hands you the formula and says use this, you'll already be comfortable with it.
