WEBVTT 00:00:06.605 --> 00:00:12.361 The next step in our study of the trig functions is to define the inverse sine, inverse cosine, and inverse tangent 00:00:12.363 --> 00:00:21.611 functions. Let's start by reviewing the graph of the sine function. You can see I've drawn it here. Its domain is 00:00:21.613 --> 00:00:32.132 all real numbers. Its range is from negative 1 to 1, and our goal is to find an inverse for this function. It'll help 00:00:32.134 --> 00:00:41.638 you a great deal to think of these x-values as angles. So your sine function takes an angle, an x-value, 00:00:41.640 --> 00:00:51.132 and spits out, or gives out, a y-value, a real number between negative 1 and 1 that is the sine of that angle. 00:00:51.134 --> 00:00:59.631 What we'd like to do is find an inverse function that takes one of these real number values between negative 1 and 1 00:00:59.633 --> 00:01:08.137 and gives us back a unique angle. But there's a big problem. Notice that if I pick say 1/2 there are a lot of angles 00:01:08.139 --> 00:01:15.640 whose sine is 1/2. It wouldn't be a unique angle. The term for this is that this is not a one-to-one function. 00:01:15.642 --> 00:01:21.184 A [non] one-to-one function doesn't have an inverse, so we have to do a little work to find an inverse function. 00:01:21.186 --> 00:01:29.353 We're going to take just a one-to-one piece of our sine function and work with that. So it might help to remember 00:01:29.355 --> 00:01:36.440 the horizontal line test from algebra. The horizontal line test, if a horizontal line crosses more than once the 00:01:36.442 --> 00:01:42.204 graph's not one-to-one. So we're looking for a little piece of the graph where when we draw that horizontal line it'll 00:01:42.206 --> 00:01:49.205 only cross one time, anywhere we draw it. There are a lot of those pieces. There's this piece. There's a piece here. 00:01:49.207 --> 00:01:59.966 There's a piece back here, and so on. It's customary to take this piece. It starts when x is negative pi over 2 00:01:59.968 --> 00:02:05.900 and ends here when x is pi over 2, and this is the piece that we're going to work on. 00:02:07.730 --> 00:02:18.230 So let's draw this one-to-one piece of the sine function separately. We'll start here. This is (negative pi over 2, -1). 00:02:18.232 --> 00:02:30.227 Here's the point (0, 0) and then the point (pi over 2, 1). And then we'll draw our sine curve. So this is our function 00:02:30.229 --> 00:02:41.258 y equals sine of x for negative pi over 2 less than or equal to x, less than or equal to pi over 2. And we can see this 00:02:41.260 --> 00:02:50.023 is our one-to-one piece we want to use. Each of these angles or x-values corresponds to one y or sine value, and 00:02:50.025 --> 00:02:59.028 furthermore, each y-value corresponds to exactly one angle value. Since we have a one-to-one piece, we're ready 00:02:59.030 --> 00:03:19.527 to define our inverse sine function. So y equals the inverse sine of x is the inverse of our function y equals sine of x 00:03:19.529 --> 00:03:31.810 for negative pi over 2 less than or equal to x, less than or equal to pi over 2. This function, y equals the inverse sine 00:03:31.812 --> 00:03:56.835 of x, has domain closed interval from negative 1 to 1 and range negative pi over 2 to pi over 2. Ok, so our 00:03:56.837 --> 00:04:05.833 inverse sine function is going to take a real number in the closed interval negative 1 to 1 and return a unique angle 00:04:05.835 --> 00:04:15.100 in the closed interval negative pi over 2 to pi over 2. Ok, one word about our notation. I know this looks like an 00:04:15.102 --> 00:04:19.611 exponent, this minus 1, but it's not an exponent. It's just a notational device. 00:04:19.613 --> 00:04:26.000 We read this "y equals the inverse sine of x." It's not an exponent. 00:04:27.800 --> 00:04:41.635 The definition of the inverse sine function tells us that the inverse sine of x is an angle. We're going to call that angle 00:04:41.637 --> 00:04:56.383 theta, in the closed interval negative pi over 2 to pi over 2, whose sine, s-i-n-e, is x. So we can write 00:04:56.385 --> 00:05:15.665 theta equals the inverse sine of x. It's just an angle, and we know that theta is in negative pi over 2 to pi over 2, 00:05:15.667 --> 00:05:23.178 this closed interval. So let's picture this here, and write this as our usual coordinate system where we look at 00:05:23.180 --> 00:05:31.678 the different quadrants. An angle of 0 radians would be drawn here, pi over 2 radians along the positive y-axis, 00:05:31.680 --> 00:05:43.174 and so on. If angle theta is in the closed interval negative pi over 2 to pi over 2, then where could theta lie? Well, 00:05:43.176 --> 00:05:53.175 here's negative pi over 2, there's pi over 2, so it can be anywhere in quadrant one or quadrant four. It could be along 00:05:53.177 --> 00:06:06.427 the positive x-axis, and it could be on either side of the y-axis, the positive or the negative. It will help us a lot 00:06:06.429 --> 00:06:15.178 to make a table. It will help us to sort of track when x is a certain number where does the terminal side of theta lie. 00:06:15.180 --> 00:06:24.985 So what are our choices? X can be negative 1. Remember our inverse sine function has domain negative 1 to 1, 00:06:24.987 --> 00:06:32.989 that closed interval, so the smallest is negative 1. Alright, well where would that be? That would mean it was along 00:06:32.991 --> 00:06:49.741 the negative y-axis, the terminal side there. In that case, we actually know the angle right away. It's going to be 00:06:49.743 --> 00:07:11.531 negative pi over 2. Anywhere in the fourth quadrant, the x must be a negative number, so we do from negative 1 to 0. 00:07:11.533 --> 00:07:21.794 Notice those are not allowing equal to, just an open interval. This is going to be in quadrant four. If x is actually 00:07:21.796 --> 00:07:44.544 equal to 0, it also corresponds to theta equals 0. That's on the positive x-axis. X might be between 0 and 1, 00:07:44.546 --> 00:07:54.545 and that's going to make the angle have its terminal side lie somewhere in quadrant one, and the last choice is 00:07:54.546 --> 00:08:08.544 x could be 1, in which case, the terminal side lies here on the positive y-axis, and we know in that case that the 00:08:08.546 --> 00:08:16.500 angle would be positive pi over 2. I think you'll find this chart very helpful as you look at these next examples. 00:08:18.500 --> 00:08:25.548 Let's practice finding exact values using our inverse sine function. The first example is to find the inverse sine 00:08:25.550 --> 00:08:33.558 of positive 1/2. So remember we're looking for an angle. We have to decide where that angle might be, and we know 00:08:33.560 --> 00:08:41.812 that its sine is positive 1/2. Ok, so in this problem, this is our x, and we can see it's positive 1/2 which is between 00:08:41.814 --> 00:08:56.825 0 and 1. So we're looking for an angle in quadrant one. In quadrant one, the angles are positive and acute, and 00:08:56.827 --> 00:09:03.830 let's look at this x-value more closely and see if it will help us figure out which special angle is the answer. 00:09:03.832 --> 00:09:14.080 The sine of pi over 6 is 1/2, so there you go. That's your answer. The positive acute angle in quadrant one whose 00:09:14.082 --> 00:09:31.080 sine is 1/2 is pi over 6. Let's try a second one. This time x is negative. It's between negative 1 and 0. So that means 00:09:31.082 --> 00:09:44.835 its terminal side is going to be somewhere in quadrant four. It needs to be a negative angle in quadrant four because, 00:09:44.837 --> 00:09:52.080 if you remember, the inverse sine function's range is negative pi over 2 to pi over 2. So we're looking for a negative 00:09:52.082 --> 00:10:02.904 angle, and I think we might want to use a reference angle to help us figure this out. Remember a reference angle is 00:10:02.906 --> 00:10:12.411 a positive acute angle. We hope that this is one of those special angles we know. The sine of pi over 4 is 00:10:12.413 --> 00:10:20.919 1 over the square root of 2, so our reference angle is pi over 4. So let's just look at the angle and make sure that we 00:10:20.919 --> 00:10:29.423 pick the right one. We want a negative angle in quadrant four. The reference angle is pi over 4, so this is actually 00:10:29.425 --> 00:10:45.675 that angle negative pi over 4. Ok, last one, inverse sine of negative 1. Negative 1 is one of these sort of special 00:10:45.677 --> 00:10:57.900 cases. The terminal side of theta will land on the negative y-axis, and theta has to be negative pi over 2. 00:11:00.290 --> 00:11:06.715 Let's turn our attention to defining the inverse cosine function. As you look at the graph of our cosine function, 00:11:06.717 --> 00:11:13.963 we see that again it is not a one-to-one function, so we need to find a one-to-one piece and define our inverse on 00:11:13.965 --> 00:11:23.464 that piece. Again we have many choices, but the customary choice is to take this first piece from 0 to pi, and it's first 00:11:23.466 --> 00:11:38.995 from the origin so that makes sense. We'll draw that one-to-one piece here and examine it more closely. 00:11:38.997 --> 00:12:00.513 We have the point (0, 1), (pi over 2, 0), and this one is (pi, -1). We're going to draw our cosine curve here. This is the 00:12:00.515 --> 00:12:08.267 function y equals cosine x for 0 less than or equal to x, less than or equal to pi, and we see that it is in fact a 00:12:08.269 --> 00:12:21.538 one-to-one function. We're ready to write the definition of the inverse cosine. So y equals the inverse cosine of x is the 00:12:21.540 --> 00:12:44.289 inverse of the function y equals cosine x on the interval from 0 to pi, the closed interval. The inverse cosine function 00:12:44.291 --> 00:13:01.287 has domain negative 1 to 1, again a closed interval, and range the closed interval from 0 to pi. So this inverse 00:13:01.289 --> 00:13:09.286 cosine function will take in a real number from this interval and give us back an angle in the closed interval 0 to pi. 00:13:10.954 --> 00:13:21.584 Let's use our definition of the inverse cosine function to help us find exact values. The definition tells us that inverse 00:13:21.586 --> 00:13:41.616 cosine of x is an angle, which we'll call theta, in the closed interval from 0 to pi whose cosine is x. So we're going 00:13:41.618 --> 00:13:54.632 to write this again a little bit differently. Theta is, the angle theta is the inverse cosine of x, and we know 00:13:54.634 --> 00:14:04.140 that 0 is less than or equal to theta, is less than or equal to pi. So where will we find the terminal side of such an 00:14:04.142 --> 00:14:17.395 angle theta? Well, it has to be from 0 to pi. Zero to pi, that's this part, quadrant one, quadrant two, the positive x-axis, 00:14:17.397 --> 00:14:30.417 the positive y-axis, or the negative x-axis. Ok, let's organize the information that we have. What are the possible 00:14:30.419 --> 00:14:38.914 values of x? Well, what's the domain of our inverse cosine function? The closed interval from negative 1 to 1. 00:14:38.916 --> 00:15:00.419 So if we have x equal to negative 1, then the terminal side will lie on the negative x-axis, and that is the angle 00:15:00.421 --> 00:15:14.460 theta equals pi. X can be something between negative 1 and 0. So that's a negative number that's going to lie, 00:15:14.462 --> 00:15:26.460 that means the terminal side will lie in quadrant two. X could be 0, in which case the terminal side lies on the 00:15:26.462 --> 00:15:45.210 positive y-axis, and so that means the angle theta is pi over 2. We have two more possibilities. If x is between 0 and 1, 00:15:45.212 --> 00:15:57.752 then the terminal side is somewhere in quadrant one. Or x could be 1, and that means that the angle lies on the 00:15:57.754 --> 00:16:12.250 positive x-axis, and that would be theta equals 0. 00:16:14.088 --> 00:16:21.528 Let's look at some examples of finding exact values of the inverse cosine function. Here's our first one. Find the 00:16:21.530 --> 00:16:32.776 inverse cosine of 1 over the square root of 2. So our x-value is 1 over the square root of 2. This is between 0 and 1, 00:16:32.778 --> 00:16:39.532 so we know that we're looking for an angle in quadrant one. In quadrant one, our angles are acute positive angles, 00:16:39.534 --> 00:16:48.027 and we just need to try to figure out if this is one of our special angles that we know. That's right. Cosine of pi over 4 00:16:48.029 --> 00:16:56.030 is 1 over the square root of 2, so the angle in the first quadrant whose cosine is 1 over the square root of 2 is 00:16:56.032 --> 00:17:08.025 pi over 4. Ok, second one, the inverse cosine of negative 1/2. So in this case, the x-value is negative 1/2. That's 00:17:08.027 --> 00:17:16.784 going to be in this interval, negative 1 to 0. That means the terminal side of our angle theta is in quadrant two. 00:17:16.786 --> 00:17:28.034 Ok, so we need to figure out which angle that is in quadrant two. We're going to use our reference angle to help us 00:17:28.036 --> 00:17:40.027 figure that out. The reference angle is a positive acute angle whose cosine is 1/2. Right, the reference angle should 00:17:40.029 --> 00:17:52.620 be pi over 3. Let's draw a picture of the angle that is going to be our answer. It's in quadrant two, the reference angle 00:17:52.622 --> 00:18:03.631 is pi over 3, and here's the angle we need. How do we figure this out? It's pi minus our reference angle, 00:18:03.633 --> 00:18:18.140 or 2 pi over 3. Ok, one more, the inverse cosine of 0. When x is 0, then we know that the terminal side of theta 00:18:18.142 --> 00:18:27.148 lies on the positive y-axis, and that makes our angle theta pi over 2. 00:18:28.520 --> 00:18:36.514 Let's now try to define an inverse tangent function. So the graph of the tangent function is shown on the board 00:18:36.516 --> 00:18:42.417 and we see, again, that it is not a one-to-one function, so in order to find an inverse we need to choose 00:18:42.419 --> 00:18:50.419 a one-to-one piece. Actually each of the pieces is one-to-one. Any will do, but the customary choice is this piece 00:18:50.421 --> 00:18:58.184 that runs from negative pi over 2 to pi over 2. Notice that at the ends those are vertical asymptotes, so we're going to 00:18:58.186 --> 00:19:07.431 choose the open interval from negative pi over 2 to pi over 2 to restrict the x's to. We'll highlight it here, and now let's 00:19:07.433 --> 00:19:27.184 draw it separately so we can look at it carefully. We have vertical asymptotes at pi over 2 and at negative pi over 2, 00:19:27.186 --> 00:19:43.228 and then we just have this one piece. This is definitely a one-to-one piece of the tangent function, and we've 00:19:43.230 --> 00:19:49.486 restricted the domain to this open interval to get a one-to-one piece. We're ready to write the definition of the 00:19:49.488 --> 00:20:11.756 inverse tangent function. Ok, y equals the inverse tangent of x is the inverse of y equals tangent x on the open interval 00:20:11.758 --> 00:20:38.785 negative pi over 2 to pi over 2. This inverse tangent function has domain all real numbers and range the open 00:20:38.787 --> 00:20:45.000 interval from negative pi over 2 to pi over 2. 00:20:46.791 --> 00:20:55.306 Let's talk about using the inverse tangent function to find exact values. From our definition, the inverse tangent of 00:20:55.308 --> 00:21:10.050 x is an angle, which we'll call theta, that is in the interval from negative pi over 2 to pi over 2, an open interval, whose 00:21:10.052 --> 00:21:29.337 tangent is x. We can write this a different way. The angle theta is inverse tangent of x. So again, this is just an angle, 00:21:29.339 --> 00:21:45.836 and we know that negative pi over 2 is less than theta which is also less than pi over 2. So let's look at where we find 00:21:45.838 --> 00:21:57.863 angles, theta, that meet this criteria. So negative pi over 2 to pi over 2 is here on the right side of this. It includes 00:21:57.865 --> 00:22:07.116 the first quadrant and the fourth quadrant. It includes the positive x-axis, and that's it because remember this is an 00:22:07.118 --> 00:22:15.629 open interval and the tangent is actually undefined at both pi over 2 and negative pi over 2. So we'll organize our 00:22:15.631 --> 00:22:32.638 information in a chart. If x is a negative number, then the terminal side of theta will lie in quadrant four, here. 00:22:32.640 --> 00:22:58.926 If x is equal to 0, then the terminal side of theta will lie on the positive x-axis, and that means that theta is 0. The 00:22:58.928 --> 00:23:11.924 only other possibility is that x is a positive number, and that would place the terminal side of theta in quadrant one. 00:23:14.000 --> 00:23:21.425 Let's look at some specific examples of finding exact values of the inverse tangent function. So our first one, 00:23:21.427 --> 00:23:30.182 inverse tangent of negative 1 over the square root of 3. So x here is a negative number, and whenever x is a negative 00:23:30.184 --> 00:23:43.426 number, the terminal side of the angle theta we're trying to find will be in quadrant four. Ok. Right, that's the tangent 00:23:43.428 --> 00:23:53.422 there of one of our special angles. So let's see if we can find a reference angle. Ok, the tangent of pi over 6 is 00:23:53.424 --> 00:24:00.739 1 over the square root of 3. So our reference angle is pi over 6, and we'll sketch a picture to make sure that we give 00:24:00.741 --> 00:24:12.997 the right name of our angle. Our angle, theta, is a negative angle in quadrant four, and the reference angle is pi over 6. 00:24:12.999 --> 00:24:28.512 So here is the angle we need. That's right. It's negative pi over 6. Ok, the inverse tangent of 1. So we look at our 00:24:28.514 --> 00:24:35.769 chart to help us out. 1 is a positive number, so that tells me the terminal side of theta is going to land somewhere in 00:24:35.771 --> 00:24:54.284 quadrant one. In quadrant one, what's the special angle whose tangent is 1? Right, pi over 4, and we have one 00:24:54.286 --> 00:25:06.040 more, inverse tangent of 0. Whenever x is 0, the terminal side of theta lies on the positive x-axis, so the angle is 0. 00:25:08.792 --> 00:25:15.041 Let's look at the instructions for this last group of examples. Use a calculator to approximate each of the following. 00:25:15.043 --> 00:25:21.319 Give your answer in radians. Round to four decimal places as needed. Well, I've already worked these out with 00:25:21.321 --> 00:25:28.316 my calculator, but I'd like to talk to you about some helpful tips. The first thing is notice that you don't really have any 00:25:28.318 --> 00:25:32.819 other way to work this problem except for the calculator approximation because this doesn't remind me of any 00:25:32.821 --> 00:25:39.834 of the special angles that we know. Ok, so how do we get this number? Well, this is important. Our answer should be 00:25:39.836 --> 00:25:47.838 in radians, so make sure your calculator is in radian mode. Where's the inverse cosine button? Usually it's the 00:25:47.840 --> 00:25:56.841 second button on the cosine key. If you can't find it, be sure to look in your calculator instruction manual. Type in 00:25:56.843 --> 00:26:03.360 inverse cosine of 4/5 and then look at the answer on the screen. Round it off to four decimal places if needed, 00:26:03.362 --> 00:26:10.360 and that's your answer. I've given you a couple of other ones to try. Make sure that you practice, that you learn 00:26:10.362 --> 00:26:16.500 how to use your calculator, and that you can get these rounded answers.