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Cofunctions definitions and examples

In mathematics, a cofunction is a function that is the inverse of another function. For example, the sine and cosine functions are cofunctions of each other, as are the logarithm and exponential functions.


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Video Transcript

Hey, guys, Mr. Clamber here. We're gonna talk about co functions. The other trig functions. Um, you might know, Heard of sine cosine tangent. And once you've gone and you know, experience and you know are well past geometry, you'll probably remember the most sine cosine and tangent. But there are other trig functions, actually, lots about the trig functions that exist. But there are only somebody there useful or common. And sine cosine intention are the three major uhh most coming trick functions. But there are three others, and they're essentially reciprocal of the true functions that we know given a right triangle. We know. I'm gonna call this data we mentioned before. This is the opposite side. This is the adjacent side on. This is the iPod news. Recall the three treat functions. Sign of theta opposite overwrite. Part news co sign of data is adjacent over high pot news and the tangent of data. Where's opposite? Over adjacent. Well, remember that trig functions are just defined ratios of a the sides of a triangle. So, for example, when we talk about the sign of an angle, I'm literally talking about the ratio of the opposite side divided by three hypotheses. It's one length divided by the other. It's purely definitional, So there are actually three other definitions. If you reciprocate all three of these, and I reciprocate, I mean the reciprocal of a over B is be over a. You literally just flip flop the numerator and denominator you turn. The fraction over the reciprocal of Sign is this function called co seek it, which is abbreviated CSC of data. It's to find as Hi pot nous over opposite. It's literally defined keyword defined to be the ratio of the high pot news length over the opposite side length. Okay, if you look at the cosine, it's reciprocal function is called. Second, it's abbreviated S E C. Secretive Fada, and that's obviously then the high pot news over do you adjacent. And then you have the reciprocal of tangent, which is called co Tangent, which is abbreviated Coty of data. And it's equal to V adjacent over the opposite. So literally you take the three trig ratios that you knew before Sankoh sine and tangent reciprocate them, which means flip them over and you have three new definitions. That's all they are definitions, which means we define them to be so co second, the reciprocal of sign. That's high partners over opposite, seeking to the reciprocal of cosign that's high partners over adjacent and then co tangent the reciprocal of adjacent over opposite. So this is a pretty much a matter of memorization. If you know the sine cosine and tangent ratios, then you technically no, the coast deacon seek it and co tangent ratios because they're literally just the reciprocal of those three. And so if we look at a couple of examples, we can solve triangles or find ratios of an angle using these definitions. So, for example, we're gonna find the seeking of theta. Remember, Sequined Pardon me sickens the reciprocal of cosine. So if the cosine of theta is theater Jace int over by pot news than what we want in terms of the secret is the high pot news over the adjacent. And so the sequence of theta in this case is 17 over 15. Look, at example three. The co tangent. It's the reciprocal of tangent. Tangent of data is opposite over adjacent. So what we want is the flip of that. We want the adjacent So in reference to this angle. This is the adjacent That's the opposite. The co attention is going to be four thirds. Also, get a co seeking situation. The co second. Okay, it's reciprocal is sign and sign. A theta is opposite over hi pot news. So in reference to this angle, this is the opposite. This the hype, a tennis. So the coast secret of theta in this case would be 17 over 15. A few things to note because the high pot news is the longest side of the triangle. All the sine and cosine ratios had to have been less than one because it was always in adjacent or opposite side divided by the hype oddness, which is the longest side. And therefore you have fractions that are less than one. The flip of that happens here, seeking and co seeking are always going to be one or greater because you're taking your high pot is which is the longest side and dividing by a leg, which is ordinarily shorter than that. The tangent coach agents can vary because it has nothing to do with the iPod news. Okay, so co tangent of theta. Remember, this is the reciprocal of tangent tangent is typically opposite over adjacent. Okay, now flip that. We're going to go. And I wrote that wrong whole lunch. So typically, um, no, I have that right. This is the opposite. This is the adjacent That's tangent. Co change is going to be the adjacent over the opposite, which is gonna be 12/24 which reduces down to, in this case, one half. If you get reduced, you typically want to. Okay, so here are a few maybe mawr advanced questions that you might be asked, um, in reference to the co functions. And that is, if you know something about sine cosine or tangent or, for that matter, any one of the six trip functions that we're talking about. Can you figure out something else about any other of the other trip functions this might require Just drawing a quick triangle, a quick sketch, labeling it and then maybe using the Pythagorean theorem and going from there. So let's say we wanted to find the coast second of fatal, which is some angle given that we know the tangent up there. Just 3/4. Well, I'm gonna draw a triangle. I'm gonna label the state it is a matter which angle you pick tangents up over adjacent. So that would be This is three. This is four. In previous videos, I've said that 345 is a Pythagorean triple. Or you can use the Pythagorean theorem to do three squared plus four squared equals pot, new squared and software it. But you're gonna find out that five is the high partners. Now, what I want is the co sequent of data. What co sequent is the reciprocal of Sign sign is opposite overhype a tennis. We want the flip of that which is hype a tennis over opposite. The answer here would be five thirds. So we drew a triangle based upon what we were given. We use Pythagorean theorem or knowledge of Pythagorean triples to figure out the missing side. And then we use our new definitions to figure out our answer. So let's try another problem. Okay, Number 26. Find the coat tangent of theta. If seeking a fe equals two. Well, first of all, this really means to over one. Think about it. It's gotta be a ratio of two sides. So if you're too, it means it's to over one draw a picture label. This data remember Sequent of Theta is the high pot noose over the adjacent because it's the reciprocal of Kassian, so that would mean the hypotenuse is too. And the adjacent side is one doing a little Pythagorean theorem. I call this, Let's say a than one squared, plus a squared. Better be two squared. That's one plus a squared equals four. That's a squared equals three. Or, in other words, a equals radical three. So we know the site is radical. Three. The co tangent, which is the reciprocal of tangent tangent, is typically the opposite over the adjacent. Now we're gonna dio adjacent over opposite. So that would be one over route three. And if you want to rationalize it, been multiplying by Route three of the Route three. You certainly could. Let's try one more of these. Question 30 here. So it says, find co attention of Fada of Sign of Fada is 13 12 13th So again, I'm gonna draw a triangle label. This data on my right angle sign is opposite over iPod news. So the opposite then will be 12. The High partners is 13 unknown. Pythagorean triple It is 5, 12 13. Check me on it. Five squared plus 12 squared equals 13 squared. I think I mentioned in a previous video that 5 12, 13 is worth memorizing. So now if I want to find the co tangent of data, this is defined to be adjacent over opposite, which is the reciprocal of tangent. The adjacent to theta is five. The opposite is 12. So the answer here is going to be five 12th. Okay, so that's essentially how you can use your new definitions of these three new trig functions, which were essentially this reciprocal of the three. We've learned to find the missing sides. And I will add this if you have a triangle and let's say you don't know theta and you know the opposite side is 12 and the adjacent site is, let's say five. And I wanted to find this angle. We've looked we've seen in previous videos that the tangent of theta equals 12/5. Or, in other words, theta equals what we call inverse tangent of 12 5th. And then you'd use your calculator to figure out that angle out. Well, we could have also done this. What's gonna get us the exact same answer as I could have said. All right, the co tangent of Fada is 5 12 because by definition, the reciprocal. So then I can say Fada must be equal to the inverse co tangent of 5/12. Thes look different, but based upon their ratios, they will generate the same angle value because they're defined to be reciprocal so that, you know, inverting, inverting the ratios air flipped. You will get the same answer using your calculator or some kind of table. So solving a triangle, finding a missing side, finding a missing angle. All that's literally the same. You just have three brand new definitions based upon three definitions we already knew. So I hope this was helpful. I will talk to you.