WEBVTT 00:00:06.121 --> 00:00:14.196 Let's look at compositions of the trig functions and their inverses. Sometimes a trig function and its inverse 00:00:14.198 --> 00:00:23.195 undo each other. Here are the six relationships we need to know. The sine of the inverse sine of x is x, as long as 00:00:23.197 --> 00:00:33.195 x is in the closed interval negative 1 to 1. Notice I switched the order, so the inverse sine of the sine of theta is theta, 00:00:33.197 --> 00:00:42.947 as long as theta is in the closed interval negative pi over 2 to pi over 2. We have similar relationships for cosine 00:00:42.949 --> 00:00:49.813 and inverse cosine compositions, and the same thing with tangent or inverse tangent compositions. The key here is 00:00:49.815 --> 00:00:58.653 the relationship is true as long as your x is in the proper interval. So you need to memorize and learn all of these 00:00:58.655 --> 00:01:08.906 relationships. Let's try some examples. The first one, the cosine of the inverse cosine of 4/5. So we need to look 00:01:08.908 --> 00:01:20.927 at 4/5 right there. In order to use this relationship, the cosine of the inverse cosine of 4/5, the answer is just 4/5 00:01:21.180 --> 00:01:30.679 as long as 4/5 is in the interval negative 1 to 1. Is it? Yes it is. It's a little bit less than 1. 00:01:30.681 --> 00:01:40.588 So this is just 4/5. Let's try a second one. The inverse cosine of the cosine of negative pi over 4. 00:01:40.590 --> 00:01:49.580 So we find the relationship we want to use. Inverse cosine of the cosine of theta is theta as long as theta is in 00:01:49.582 --> 00:01:56.837 the interval 0 to pi. Well, it's not. Negative pi over 4 is definitely not in this interval. So what does that mean? 00:01:56.839 --> 00:02:05.837 It doesn't mean we can't answer the question. It just means that we have to work a little harder to find that answer. 00:02:05.839 --> 00:02:13.839 So let's start with this part, the cosine of negative pi over 4. We already know how to find this. Negative pi over 4, 00:02:13.841 --> 00:02:25.952 that's in the fourth quadrant. The cosine of an angle in the fourth quadrant is positive. So this inside part is just 00:02:25.954 --> 00:02:41.096 1 over the square root of 2, positive cosine of pi over 4. By evaluating the inside part, it now reduces to a different 00:02:41.098 --> 00:02:48.436 kind of problem, and we know how to solve this as well. The inverse cosine of 1 over the square root of 2. 00:02:48.438 --> 00:02:59.529 This is a positive number between 0 and 1, so in what quadrant will the terminal of this angle lie? Right, 00:02:59.531 --> 00:03:15.500 quadrant one. So 1 over the square root of 2 is positive, and so this is going to lie in quadrant one. What angle is it? 00:03:15.502 --> 00:03:28.303 Well, it's in quadrant one, so it's a positive acute angle, and its cosine is 1 over the square root of 2, so pi over 4. 00:03:28.305 --> 00:03:38.645 We have one more. The inverse cosine of the cosine of 2 pi over 3. Alright, so we go back and we look at our 00:03:38.647 --> 00:03:49.329 relationship. The inverse cosine of the cosine of theta is theta as long as theta is in the closed interval 0 to pi. So 00:03:49.331 --> 00:03:59.080 we need to know is 2 pi over 3 between 0 and pi. Yes, it is. It's somewhere in the second quadrant. That's correct. 00:04:02.000 --> 00:04:10.762 Here are a few more examples. The inverse sine of the sine of 2 pi over 7. So we find the relationship that works 00:04:10.764 --> 00:04:20.010 here, or at least we hope it works. The inverse sine of the sine of theta is theta for any theta in the closed interval 00:04:20.012 --> 00:04:31.266 from negative pi over 2 to pi over 2. So the big question here is, is 2 pi over 7 in that interval, negative pi over 2 00:04:31.268 --> 00:04:47.718 to pi over 2? Ok, so let's think about 2 pi over 7. It's a little bit less than pi over 2, half of pi, so yes, it's right there. 00:04:47.720 --> 00:04:57.470 It's somewhere in this interval. Since the answer is yes, then I'm allowed to use this relationship. The inverse sine 00:04:57.472 --> 00:05:10.721 of the sine of 2 pi over 7 is 2 pi over 7. Alright, let's look at the next one. The sine of the inverse sine of 3/2. 00:05:10.723 --> 00:05:23.086 The sine of the inverse sine of 3/2, and I see a problem already. 3/2 is too big. It's not in negative 1 to 1, 00:05:23.088 --> 00:05:29.179 that interval. So I can't use this relationship. Again, I have to stop and think about whether I can even solve 00:05:29.181 --> 00:05:41.928 the problem. This number is an angle whose sine is 3/2. Right, that's not even possible, so this expression 00:05:41.930 --> 00:06:00.182 is undefined. Let's look at one more. The inverse sine of the sine of 2 pi over 3. So here's our relationship. 00:06:00.184 --> 00:06:08.429 Inverse sine of the sine of 2 pi over 3 would be 2 pi over 3, as long as 2 pi over 3 is in this closed interval 00:06:08.431 --> 00:06:16.928 negative pi over 2 to pi over 2. But correct, it's not in that interval. 2 pi over 3 is too big. Well, when the relationship 00:06:16.930 --> 00:06:22.931 doesn't hold we have to stop and think if we can work it out in some other way using other infomation we know. 00:06:22.933 --> 00:06:32.818 So let's look at the inside part here, the sine of 2 pi over 3. In what quadrant does 2 pi over 3's terminal side lie? 00:06:32.820 --> 00:06:51.325 Right, quadrant two. We know that in quadrant two the sine of an angle is positive or negative? Right, positive. 00:06:51.327 --> 00:06:58.687 We could use a reference angle to help a little bit. The reference angle for 2 pi over 3 would be pi over 3, 00:06:58.689 --> 00:07:10.435 and we know the sine of pi over 3 is the square root of 3 over 2. Ok, so we fill in just this inside part. We've 00:07:10.437 --> 00:07:19.618 discovered that the sine of 2 pi over 3 is positive square root of 3 over 2, using this information and our 00:07:19.620 --> 00:07:26.621 reference angle. We have one more step to finish the problem, the inverse sine of square root of 3 over 2. 00:07:26.623 --> 00:07:35.460 Ok, inverse sine of this number is an angle. Where does the terminal side of this angle lie? Correct, this is positive, 00:07:35.462 --> 00:07:48.500 so it's going to lie in quadrant one, and the angle in quadrant one whose sine is square root of 3 over 2 is pi over 3. 00:07:51.000 --> 00:08:00.210 Let's look at another set of examples. We'd like to find the tangent of the inverse tangent of 15. So the tangent of 00:08:00.212 --> 00:08:09.459 the inverse tangent of x equals x for any x. This is my favorite one because it's going to have to just be 15. 00:08:09.461 --> 00:08:19.957 The second one, the inverse tangent of the tangent of negative pi over 3. This relationship, the inverse tangent 00:08:19.959 --> 00:08:28.015 of the tangent of an angle is that angle as long as the angle is in this open interval, negative pi over 2 to pi over 2. 00:08:28.017 --> 00:08:39.353 So where will I find negative pi over 3? Inside this interval? Yes, so this one is negative pi over 3. 00:08:39.355 --> 00:08:51.609 The inverse tangent of the tangent of 7 pi over 12. So this would just be 7 pi over 12, if 7 pi over 12 is in this 00:08:51.611 --> 00:09:01.107 interval, negative pi over 2 to pi over 2. Is it? Unfortunately no, 7 pi over 12 is a little bit bigger than 6 pi over 12, 00:09:01.109 --> 00:09:08.603 so it's a little bit bigger than pi over 2. This isn't going to work. I have to do something else to find the answer. 00:09:08.605 --> 00:09:14.614 Well, there's another problem with this one. It's not one of your special angles that you know and love, right? Ok, 00:09:14.616 --> 00:09:24.416 so we're going to draw some pictures and see if we can figure it out. We said 7 pi over 12 is a little bit bigger 00:09:24.418 --> 00:09:38.915 than pi over 2, so 7 pi over 12 has its terminal side in quadrant two. Well, I know a couple of things about angles 00:09:38.915 --> 00:09:47.666 in quadrant two. For instance, would the tangent of 7 pi over 12 be a positive number or a negative number? 00:09:47.668 --> 00:09:57.180 A negative number, so that might be useful information in a few minutes. I'm just going to make a note of that 00:09:57.182 --> 00:10:06.700 for now. What about the reference angle? Can we figure out the reference angle? We'll call that theta r. Do you 00:10:06.702 --> 00:10:15.452 remember how to find a reference angle? All the way to here would be pi, so we're going to find that theta r is 00:10:15.454 --> 00:10:30.479 pi minus 7 pi over 12, and that means theta r is 5 pi over 12. Now what? We're going to try to find the 00:10:30.481 --> 00:10:40.478 inverse tangent of whatever this number is, and we know that it's a negative number. So let's think about our 00:10:40.478 --> 00:10:48.000 inverse tangent function. If you recall, the inverse tangent function has to give you an angle over here, between 00:10:48.002 --> 00:10:59.260 negative pi over 2 and pi over 2. If this is a negative number, then it must be in quadrant four. So our answer 00:10:59.262 --> 00:11:08.515 is somewhere down here in quadrant four. I'm going to draw that in purple on this one. We want an angle whose 00:11:08.517 --> 00:11:24.785 terminal side is in quadrant four. The reference angle is going to have to match this one, 5 pi over 12. 00:11:24.787 --> 00:11:31.787 So if I draw that negative angle in the fourth quadrant, the same reference angle as this one, then I find out 00:11:31.789 --> 00:11:41.000 that my angle is negative 5 pi over 12. 00:11:42.536 --> 00:11:49.309 In this next example, notice that we're looking for the tangent of the inverse sine of 1 over the square root of 2. 00:11:49.311 --> 00:11:55.314 So I've sort of mixed up our trig functions and inverse trig functions. None of those special relationships will help us, 00:11:55.316 --> 00:12:03.074 and we simply need to work this out starting with the inside part. So let's start with the inverse sine of 00:12:03.076 --> 00:12:12.577 1 over the square root of 2. This is an angle. In what quadrant does this angle's terminal side lie? Well, since this is 00:12:12.579 --> 00:12:19.081 a positive number and this is the inverse sine function, it must be in quadrant one. This is an angle in quadrant one 00:12:19.083 --> 00:12:37.359 whose sine is 1 over the square root of 2. You already know what that angle is, don't you? It's pi over 4. 00:12:37.361 --> 00:12:47.370 If we make this substitution, this is just pi over 4. We've simplified this to the question just asking what's the 00:12:47.372 --> 00:12:53.872 tangent of pi over 4, and that's 1. 00:12:54.500 --> 00:13:00.619 Here we'd like to find the secant of the inverse tangent of negative square root of 3, and again we're going to start 00:13:00.621 --> 00:13:08.640 on the inside part of this and work our way through. Inverse tangent of negative square root of 3, so this is going 00:13:08.642 --> 00:13:17.890 to be an angle whose terminal side lies in which quadrant? Ok, well this is negative and this is an inverse tangent, 00:13:17.892 --> 00:13:36.916 so quadrant four, and it's the angle in quadrant four whose tangent is negative square root of 3. So have you 00:13:36.918 --> 00:13:46.167 figured out which one it is? Tangent of pi over 3 is the square root of 3, so we want an angle in quadrant four, 00:13:46.167 --> 00:13:57.166 a negative angle, whose tangent is negative square root of 3. We're using pi over 3 as our reference angle, 00:13:57.168 --> 00:14:08.169 so this must be the angle negative pi over 3. Next question, what's the secant of negative pi over 3? 00:14:08.171 --> 00:14:15.671 I think it'll be easier for us to find the cosine of negative pi over 3 and use that to get our answer. Where is 00:14:15.671 --> 00:14:27.919 negative pi over 3's terminal side? Quadrant four. In quadrant four, is the cosine positive or negative? Positive. 00:14:27.921 --> 00:14:40.482 Ok, so it's positive, and we need to know the value. What's the cosine of pi over 3? 1/2, ok and we've already 00:14:40.484 --> 00:14:47.979 decided that it's positive so this is 1/2. Ok, that wasn't quite my answer though. Cosine of negative pi over 3 is 1/2, 00:14:47.981 --> 00:14:56.700 so what's the secant of negative pi over 3? 2 and there we go. 00:14:58.245 --> 00:15:06.005 Let's find the inverse sine of the cosine of 7 pi over 4. So again we'll start on the inside and work our way through 00:15:06.007 --> 00:15:18.767 the problem. 7 pi over 4 is an angle whose terminal side lies in what quadrant? The fourth. Let's just draw a little 00:15:18.769 --> 00:15:25.274 picture of that. It's going to be down here. 7 pi over 4 is all the way around. Is the cosine of an angle positive or 00:15:25.276 --> 00:15:34.522 negative in quadrant four? Positive, so we're going to be looking for the inverse sine of a positive number. You need 00:15:34.524 --> 00:15:42.289 to fill that in. What's the reference angle? Pi over 4, ok, so what's the cosine of pi over 4? Right, 00:15:42.291 --> 00:15:50.060 1 over the square root of 2. We've already decided that should be a positive number. Inverse sine of 00:15:50.062 --> 00:16:00.291 1 over the square root of 2. What quadrant can this angle lie in, its terminal side? The first, right. In the first 00:16:00.293 --> 00:16:09.788 quadrant what's the angle whose sine is 1 over the square root of 2? Correct, it's pi over 4. 00:16:11.530 --> 00:16:20.330 Let's find the inverse cosine of the sine of 5 pi over 3. We'll start with the inside part, the sine of 5 pi over 3. 00:16:20.332 --> 00:16:31.829 Tell me about the angle 5 pi over 3. It lands in, it's terminal side lands in, quadrant four. What would be the reference 00:16:31.831 --> 00:16:46.076 angle? Right, pi over 3. In the fourth quadrant, is the sine of an angle positive or negative? Negative. So the sine 00:16:46.078 --> 00:16:56.575 of 5 pi over 3 is negative. Let's figure that number out. We said the reference angle is pi over 3. We know that the 00:16:56.577 --> 00:17:06.827 sine of pi over 3 is square root of 3 over 2, so this is equal to negative square root of 3 over 2. So we really need 00:17:06.829 --> 00:17:16.577 to find now the inverse cosine of negative square root of 3 over 2. So let's think about our inverse cosine function. 00:17:16.579 --> 00:17:29.574 The answer here is an angle somewhere in the interval 0 to pi. Since this is a negative number, that means I'm looking for an 00:17:29.576 --> 00:17:40.826 angle whose terminal side is in quadrant two and whose cosine is negative square root of 3 over 2. A reference angle 00:17:40.828 --> 00:17:50.417 will probably help us here. What positive acute angle's cosine is square root of 3 over 2? 00:17:50.419 --> 00:18:04.420 Pi over 6. We're going to draw a picture. We know our angle is in quadrant two, that our reference angle is 00:18:04.422 --> 00:18:20.500 pi over 6, and so we know that the angle we're looking for, whose terminal side lies in quadrant two, must be 5 pi over 6. 00:18:22.453 --> 00:18:30.451 Here we'd like to find the sine of the inverse cosine of negative 7/8. We'll start on the inside. We're looking for an 00:18:30.453 --> 00:18:38.704 angle whose cosine is negative 7/8, and where does the terminal side of this angle lie? Well, it's an inverse cosine 00:18:38.706 --> 00:18:47.953 question and this number is negative. In the interval 0 to pi, that must mean we're looking in quadrant number two. 00:18:47.955 --> 00:18:59.999 So we're going to sketch the angle in quadrant two. The terminal side lies here, and we're going to label a point on 00:19:00.001 --> 00:19:08.497 the terminal side of the angle, and we're going to write down the x, the y, and the r that correspond to our picture. 00:19:08.499 --> 00:19:18.008 Since we're given the cosine of the angle is negative 7/8 and cosine is x over r, we know that x is negative 7. R has 00:19:18.010 --> 00:19:30.770 to be positive, so it's just 8, and we'll need to find y. Well, look at our picture. Y is up here, and that means that y is a 00:19:30.772 --> 00:19:43.519 positive number since it lands in quadrant two here. So y has to be the positive square root of r squared minus 00:19:43.521 --> 00:19:50.269 x squared, and when we substitute our known values and work this out we're going to find out that y is the square 00:19:50.271 --> 00:20:01.048 root of 15. Now, unfortunately, I don't know exactly what the value of this angle is, but I have everything I need to 00:20:01.050 --> 00:20:10.307 know about it in order to find the sine of that angle. Sine, remember the definition, it's y over r, so the 00:20:10.309 --> 00:20:18.310 y is square root of 15, and r was positive 8.