WEBVTT 00:00:05.605 --> 00:00:10.113 Hi. Let's take a look at an interesting property of right triangles. 00:00:10.115 --> 00:00:22.113 Suppose we draw a right triangle, and we label the sides a, b, and c. 00:00:22.115 --> 00:00:27.630 The side c, which is the side opposite the right angle, is called the hypotenuse. 00:00:27.632 --> 00:00:35.380 That's h-y-p-o-t-e-n-u-s-e. 00:00:35.382 --> 00:00:44.880 Now, in terms of the angle theta, which I will select this angle and call it theta, 00:00:44.882 --> 00:00:54.880 then the side b is the side opposite the angle theta, 00:00:54.882 --> 00:01:06.126 and the side a is the side next to, or adjacent, the angle theta. 00:01:06.128 --> 00:01:10.675 Now let's take a look at the ratios of the sides of the right triangle. 00:01:10.677 --> 00:01:18.432 Let's pick, for example, the ratio of the length of side b to the length of side a. 00:01:18.434 --> 00:01:28.934 We could write that as b divided by a. Now, here's the interesting property. 00:01:28.936 --> 00:01:42.936 If we extend this side and we extend this side (even if it goes through the words) 00:01:42.938 --> 00:01:50.185 and then we close out the right triangle, we see something interesting. 00:01:50.187 --> 00:02:02.931 We can call this side right here b prime, and then we call this new side here a prime, 00:02:02.933 --> 00:02:06.728 and here's the part that's really interesting. 00:02:06.730 --> 00:02:16.980 The ratio of b prime to a prime is the same as the ratio of b to a. 00:02:16.982 --> 00:02:23.981 So what that means is that the ratio of the sides of the triangle depend only 00:02:23.983 --> 00:02:29.481 on the specific angle theta that is selected, as long as you're working in a right triangle. 00:02:29.483 --> 00:02:37.229 So in trigonometry, the ratios of all the pairs of sides are given special names. 00:02:37.231 --> 00:02:41.764 There are six of them. Let's go through them and define them in terms of the sides 00:02:41.766 --> 00:02:49.013 of the triangle. The first one is called the sine, and it's spelled s-i-n-e. 00:02:49.015 --> 00:02:59.016 It's abbreviated s-i-n, and the way it works is this. The sine of theta is the value 00:02:59.018 --> 00:03:04.019 given to the ratio of the opposite side divided by the hypotenuse, 00:03:04.021 --> 00:03:10.543 and we'll just abbreviate here, opposite divided by hypotenuse. 00:03:10.545 --> 00:03:18.535 The next function we're going to look at is the cosine function. It is abbreviated 00:03:18.537 --> 00:03:32.785 c-o-s, and so the cosine of theta is the ratio of the adjacent side to the hypotenuse. 00:03:32.787 --> 00:03:47.002 The next one is called the tangent. It is abbreviated t-a-n, and the tangent of theta 00:03:47.004 --> 00:03:52.833 is the ratio of the opposite side to the adjacent side. 00:03:52.835 --> 00:04:00.036 Alright, that's half of them. There are three more. 00:04:00.038 --> 00:04:12.353 The next one is called the cosecant, c-o-s-e-c-a-n-t, abbreviated c-s-c, 00:04:12.355 --> 00:04:24.338 and the cosecant of theta is the ratio of the hypotenuse to the opposite side. 00:04:24.340 --> 00:04:37.588 The next one is called the secant. It is abbreviated s-e-c, and the secant of theta 00:04:37.590 --> 00:04:47.839 is the ratio of the hypotenuse to the adjacent side. 00:04:47.841 --> 00:05:01.092 And then the last one is the cotangent, abbreviated c-o-t, and it is the ratio of 00:05:01.094 --> 00:05:09.092 the adjacent side to the opposite side. 00:05:09.094 --> 00:05:22.592 These six ratios and their associated names in terms of angle theta 00:05:22.594 --> 00:05:29.587 are called the right triangle definitions of the trig functions of theta. 00:05:29.589 --> 00:05:33.937 You need to study these carefully. You need to practice writing them over and over 00:05:33.939 --> 00:05:37.175 again until you're absolutely comfortable with them. 00:05:37.177 --> 00:05:42.692 Now let's take a look at an example of how this is used. 00:05:42.694 --> 00:05:47.942 Use the definitions to find the trig functions for the given triangle. 00:05:47.944 --> 00:05:51.943 Alright, so what do we have? We have a right triangle. 00:05:51.945 --> 00:05:56.444 We have a designated angle, theta, and we know that the side 00:05:56.446 --> 00:06:08.695 opposite theta is 2 units long and the side adjacent to theta is 5 units long. 00:06:08.697 --> 00:06:13.441 So, what's missing? Well, of course, what's missing is the hypotenuse, 00:06:13.443 --> 00:06:18.728 this side right here. We know this is a right triangle, so let's find the hypotenuse. 00:06:18.730 --> 00:06:23.482 We want to know the length of side c. Alright now what's the relationship? 00:06:23.484 --> 00:06:24.225 Think about it. 00:06:24.227 --> 00:06:29.720 What does the Pythagorean theorem say about the relationships of the 3 sides? 00:06:29.722 --> 00:06:39.476 Right. It is c squared equals a squared plus b squared. We're looking for side c. 00:06:39.478 --> 00:06:48.503 So we're going to write c squared is, a is 5, so that's 5 squared, and b is 2, 00:06:48.505 --> 00:06:58.253 so that's 2 squared, so c squared is 25 plus 4 which is 29. Now, 00:06:58.255 --> 00:07:05.023 this is an algebraic equation, and so when you take the square root of both sides 00:07:05.025 --> 00:07:12.023 algebraically, unrelated to geometry, you would get plus or minus the square root of 29. 00:07:12.025 --> 00:07:15.271 But because this is the length of the side of a triangle, 00:07:15.273 --> 00:07:19.288 there's a geometric restriction here which says of course the side of a triangle 00:07:19.288 --> 00:07:27.000 can't be negative, and so in our case, c is the square root of 29. 00:07:27.002 --> 00:07:31.304 By the way, you will do this process many times in trigonometry. You may want to 00:07:31.306 --> 00:07:35.553 start trying to think of doing it mentally, and here's what you would do. 00:07:35.555 --> 00:07:40.539 You're looking for the hypotenuse. You square this side, 25, this side, 4, 00:07:40.541 --> 00:07:46.816 add them together, 29, square root of 29. So get used to practicing this mentally. 00:07:46.818 --> 00:07:49.817 It will save you a lot of time. So back to the problem. 00:07:49.819 --> 00:07:53.324 We need to use the definitions. Do you remember them? 00:07:53.324 --> 00:07:57.080 If not, get them out in front of you and then practice, so in the future 00:07:57.082 --> 00:08:01.834 you will remember the six trig functions. OK, here we go. We start with sine. 00:08:01.836 --> 00:08:11.085 The sine of theta is, say it to yourself, it is the opposite over the hypotenuse, 00:08:11.087 --> 00:08:18.584 so that's 2 over the square root of 29. OK, now we're going to stop for a second. 00:08:18.586 --> 00:08:23.854 Unless you are specifically told to do so, there is no reason why you have to 00:08:23.856 --> 00:08:28.111 rationalize this denominator. It is perfectly acceptable to leave this 00:08:28.113 --> 00:08:35.607 unless you have instructions that tell you otherwise. The cosine of theta is 00:08:35.609 --> 00:08:43.065 adjacent over hypotenuse, so that's 5 over the square root of 29. 00:08:43.067 --> 00:08:49.817 And then the tangent of theta. OK say it to yourself. What is it? 00:08:49.819 --> 00:08:57.066 It is the opposite over the adjacent, so it's 2 over 5. 00:08:57.068 --> 00:09:06.065 Alright, next 3. Cosecant of theta is the hypotenuse over the opposite, 00:09:06.067 --> 00:09:12.400 square root of 29 over 2. Be careful when you write this here. 00:09:12.402 --> 00:09:16.904 The 29 is under the radical. The division sign and the 2 are not. 00:09:16.906 --> 00:09:25.410 Next is the secant of theta, and the secant of theta is the hypotenuse over the 00:09:25.412 --> 00:09:31.919 adjacent. Correct. So it's the square root of 29 all over 5. 00:09:31.921 --> 00:09:39.667 And the last one is cotangent, and the cotangent of theta is equal to 00:09:39.669 --> 00:09:48.417 the adjacent over the opposite, and so it is 5 over 2. 00:09:48.419 --> 00:09:52.668 OK, this satisfies the directions, but let's look at what we have here. 00:09:52.670 --> 00:10:01.419 These are the six trig functions of theta for this triangle. 00:10:01.421 --> 00:10:06.668 We calculated these using the definitions, but you have probably noticed at this point 00:10:06.671 --> 00:10:10.441 that there are certain relationships between the six of these. 00:10:10.443 --> 00:10:14.460 So the next thing we're going to do is we're going to take a look at a little bit 00:10:14.462 --> 00:10:19.970 faster way to get some of these by using the relationships between them. 00:10:21.223 --> 00:10:27.473 Alright, I have here on the board the definitions, the right triangle definitions, 00:10:27.475 --> 00:10:31.231 of the six trig functions. What we want to do is look at the relationships. 00:10:31.233 --> 00:10:37.236 We're trying to see how these fit together and how one is related to the other. 00:10:37.238 --> 00:10:45.986 Well first of all, I notice that sine divided by cosine, well what would it be? 00:10:45.988 --> 00:10:49.735 It would be this expression divided by this one, and if you simplified it, 00:10:49.737 --> 00:10:52.731 hypotenuse and hypotenuse would cancel and you'd end up with 00:10:52.733 --> 00:10:56.507 opposite over adjacent which is 00:10:56.509 --> 00:11:00.500 tangent. Ok, so we notice that relationship, so we say that 00:11:00.502 --> 00:11:11.509 the sine of theta divided by the cosine of theta is equal to the tangent of theta. 00:11:11.511 --> 00:11:13.509 Now let's look at these. 00:11:13.511 --> 00:11:22.032 What is the relationship between sine, cosine, and cotangent? 00:11:22.034 --> 00:11:25.778 Obviously there are lots of relationships, but there are some that seem to be used 00:11:25.780 --> 00:11:28.531 more often than others, so those are the ones we're going to point out here. 00:11:28.533 --> 00:11:32.543 What happens if you divide cosine by sine? 00:11:32.545 --> 00:11:36.791 Well again hypotenuse cancels, but you get adjacent over opposite 00:11:36.793 --> 00:11:42.000 which is cotangent. So we're going to write down this one. 00:11:42.002 --> 00:11:50.542 Cosine of theta divided by sine of theta equals cotangent of theta. 00:11:50.544 --> 00:12:03.043 Alright, these two in particular are what are called the quotient identities, 00:12:03.045 --> 00:12:06.543 and as I said, there are others, but these are the most used. 00:12:06.545 --> 00:12:11.834 Now let's take a look at the reciprocal identities. 00:12:11.836 --> 00:12:21.333 Well, sine is opposite over hypotenuse. Cosecant is hypotenuse over opposite. 00:12:21.335 --> 00:12:34.604 So what that means is that cosecant of theta is equal to 1 over the sine of theta, 00:12:34.606 --> 00:12:39.109 and for that matter, the sine of theta is equal to 1 over the cosecant of theta. 00:12:39.111 --> 00:12:43.050 But once you've learned the identity one way, of course, you can simplify 00:12:43.052 --> 00:12:46.859 it to get the other one. What else do we see? 00:12:46.861 --> 00:12:55.109 Well we see that the secant of theta is equal to 1 over, what? 00:12:55.111 --> 00:13:02.131 Mmhm, cosine, and then, what about the cotangent? 00:13:02.133 --> 00:13:12.642 The cotangent of theta is 1 over the tangent of theta, and these very descriptively 00:13:12.644 --> 00:13:25.610 are called re-cip-ro-cal, reciprocal identities because that describes the relationship. 00:13:25.612 --> 00:13:31.392 OK, the next group of identities take a little bit more thought, but not that much. 00:13:31.394 --> 00:13:39.904 What happens if you take the sine and you square it and the cosine and you square it? 00:13:39.906 --> 00:13:44.674 Well it's not quite as obvious from these definitions, but if you're curious 00:13:44.676 --> 00:13:48.424 about where this comes from, you can go back to a previous example that we did 00:13:48.426 --> 00:13:51.181 and actually take those values we did with the triangle that had 00:13:51.183 --> 00:13:54.432 the 2 and the 5 and the square root of 29 and you can square the values, 00:13:54.434 --> 00:13:57.433 and see what happens and I'll show you the result. 00:13:57.435 --> 00:14:03.690 The result is that if you take the sine and you square it, so the sine of theta 00:14:03.692 --> 00:14:11.942 squared, and you add it to the cosine squared of theta you will get 1. 00:14:11.944 --> 00:14:24.213 Notice that this is really the same expression as the sine of theta squared. 00:14:24.215 --> 00:14:28.468 It doesn't mean anything strange like square a number and multiply it by theta. 00:14:28.468 --> 00:14:32.966 This is all one term raised to the second power, but this is how it's written. 00:14:32.968 --> 00:14:39.714 The next relationship that follows this pattern is that if you square the tangent 00:14:39.716 --> 00:14:47.216 of theta and you add 1 you will get the value of the secant squared of theta, 00:14:47.218 --> 00:14:52.993 and the third one is that if you take the cotangent of theta and square it 00:14:52.995 --> 00:14:58.995 and add 1 you will get the cosecant squared of theta. 00:14:58.997 --> 00:15:10.261 These are called the Pythagorean identities. 00:15:10.263 --> 00:15:23.262 Together this group, this whole group together, these are called 00:15:23.264 --> 00:15:28.278 the fundamental identities. You need to practice these. 00:15:28.280 --> 00:15:31.778 You need to be very familiar with them. You need to be able to write them down. 00:15:31.780 --> 00:15:36.776 You need to be able to recall them with barely any effort whatsoever. 00:15:36.778 --> 00:15:40.530 Now let's take a look at how this can be used. 00:15:42.791 --> 00:15:48.041 Example 2: Use identities to find the exact value of, 00:15:48.043 --> 00:15:52.293 and we have two expressions. Now down here at the bottom of the board 00:15:52.295 --> 00:15:58.543 I have the fundamental identities that we developed in the last segment. 00:15:58.545 --> 00:16:01.540 I'm going to just leave them here so that we can refer to them. 00:16:01.542 --> 00:16:05.313 So let's see what we have. We have the tangent of 75 minus, 00:16:05.315 --> 00:16:10.066 and then we have the sine of 75 degrees divided by the cosine of 75 degrees. 00:16:10.068 --> 00:16:14.000 OK, so what's the identity that we can use here to simplify this? 00:16:14.002 --> 00:16:18.073 Well, sine of theta divided by cosine of theta is tangent of theta, 00:16:18.075 --> 00:16:21.321 and since these angles are the same, both 75 degrees, 00:16:21.323 --> 00:16:27.574 we can rewrite this expression by copying over the first term, 00:16:27.576 --> 00:16:35.075 making no changes, and then replacing the second expression with 00:16:35.077 --> 00:16:42.824 the tangent of 75, and at this point it's pretty obvious that the exact value is 0. 00:16:42.826 --> 00:16:48.857 Alright, let's look at the second one. As soon as you see the 00:16:48.859 --> 00:16:55.608 sine squared pi over 5 and then secant squared pi over 5, where are you going to go? 00:16:55.610 --> 00:16:59.619 Sure, you're going to go to these identities here. Now we have sine squared 00:16:59.621 --> 00:17:05.121 and secant squared so that is a possibility of the first one or the second one. 00:17:05.123 --> 00:17:08.617 Let's see. Which one has the most promise? 00:17:08.619 --> 00:17:15.640 Well, we know that secant is related to what? Look around. 00:17:15.642 --> 00:17:20.641 Secant is related to cosine. They're reciprocals, 00:17:20.643 --> 00:17:26.640 and if I can change a secant expression to be defined in terms of cosine, 00:17:26.642 --> 00:17:30.154 then I should be able to use this identity second. 00:17:30.156 --> 00:17:39.159 Alright, so the first thing I'm going to do is copy the first term, and then 00:17:39.161 --> 00:17:43.910 I'm going to rewrite the secant squared pi over 5. Let's see. 00:17:43.912 --> 00:17:48.664 Secant and cosine are reciprocals, so if I have 1 over the secant squared 00:17:48.667 --> 00:17:56.658 of an angle, I should be able to write it as the cosine squared of the same angle. 00:17:56.658 --> 00:18:00.930 Now, this is actually the reverse of the identity that I have written here. 00:18:00.932 --> 00:18:06.678 I'm actually using the cosine of theta is 1 over the secant of theta. But that's fine. 00:18:06.678 --> 00:18:09.442 Once you know this one you can get the other one. And then what do I have? 00:18:09.444 --> 00:18:15.697 I have sine squared of an angle plus cosine squared of an angle is 1. 00:18:15.699 --> 00:18:19.452 Alright, let's look at another example of how we use these identities. 00:18:22.200 --> 00:18:28.701 Example 3: Given that the sine of theta is 1/4 and that theta is an acute angle, 00:18:28.703 --> 00:18:32.953 find the exact value of the remaining five trig functions. 00:18:32.955 --> 00:18:37.702 Alright, so let's take a look at what we have. So we have a right triangle. 00:18:37.704 --> 00:18:40.466 We know that because we are using trig functions, and of course, 00:18:40.468 --> 00:18:45.213 they are trig functions of right triangles. So we have a right triangle. 00:18:45.215 --> 00:18:48.966 Let me just draw a little bitty one right here. Just enough to label the sides. 00:18:48.968 --> 00:18:55.468 Here's angle theta, and the sine is 1 divided by 4. Think about the definition again. 00:18:55.470 --> 00:19:02.990 So that means, 1, so that means this side is 1, and the hypotenuse is 4. 00:19:02.992 --> 00:19:06.491 So what's missing is this side. Now remember how I told you to 00:19:06.493 --> 00:19:09.739 learn how to do this mentally? Let's see if we can find this side mentally. 00:19:09.741 --> 00:19:13.743 It would be c squared equals a squared plus b squared. 00:19:13.745 --> 00:19:18.757 So what that means is this side squared is that one squared minus that one squared. 00:19:18.759 --> 00:19:20.008 So what do we get? 00:19:20.010 --> 00:19:27.758 We get 16 minus 1 which is 15, which means this side is the square root of 15 units. 00:19:27.760 --> 00:19:31.090 16 minus 1 and then square root of both sides. 00:19:31.092 --> 00:19:35.007 OK, so what we have is opposite, adjacent, and hypotenuse. 00:19:35.009 --> 00:19:44.533 But, we have all of this. So, we start. The sine of theta is given. 00:19:44.535 --> 00:19:49.530 Notice how every time I do this I write three down, three more down, in the same order. 00:19:49.532 --> 00:19:53.033 It helps you to remember these things if you are very orderly and organized 00:19:53.035 --> 00:20:00.033 about how you do this. Now, the cosine of theta. It is going to be 00:20:00.035 --> 00:20:13.283 adjacent over hypotenuse. Now, use the identities. What is tangent? 00:20:13.285 --> 00:20:18.067 Tangent is cosine divided by sine, and the 4's will cancel, 00:20:18.069 --> 00:20:23.066 and what you get is 1 over the square root of 15. Oops, let's look at that. 00:20:23.068 --> 00:20:30.814 Tangent, I think I said cosine over sine. Ooo, no. Tangent is sine over cosine. 00:20:30.816 --> 00:20:38.337 Alright, so the tangent of theta, let's look at it again, is sine divided by cosine. 00:20:38.339 --> 00:20:40.836 And you do have to be careful when you do this because of course 00:20:40.838 --> 00:20:42.338 it's the same words over and over again. 00:20:42.340 --> 00:20:46.344 So it would be 1/4 divided by the square root of 15 fourths, the 4's cancel, 00:20:46.346 --> 00:20:52.094 and so what you do is write 1 over the square root of 15. 00:20:52.096 --> 00:20:57.591 OK, now basically at this point you've got all of them because the next three are 00:20:57.593 --> 00:21:03.094 reciprocals of each of these three. So as long as you are careful with what you write, 00:21:03.096 --> 00:21:10.093 the cosecant of theta would be the reciprocal of 1/4 which is 4. 00:21:10.095 --> 00:21:20.090 The secant of theta is the reciprocal of the cosine, 00:21:20.092 --> 00:21:28.594 and the cotangent of theta is the reciprocal of the tangent. 00:21:28.596 --> 00:21:32.590 Alright now, before you stop, you go back and you check all of this. 00:21:32.592 --> 00:21:34.340 This was given, 00:21:34.342 --> 00:21:37.094 cosine is this over this, 00:21:37.096 --> 00:21:41.593 tangent is that divided by that, from this identity. 00:21:41.595 --> 00:21:44.844 Cosecant is the reciprocal of this one, 00:21:44.846 --> 00:21:47.094 secant the reciprocal of this one, 00:21:47.096 --> 00:21:51.344 cotangent the reciprocal of this one, and you've got all six. 00:21:51.346 --> 00:21:55.596 Let's look at another property of right triangles. 00:21:57.668 --> 00:22:02.913 Let's discuss the relationship of cofunctions of complementary angles. 00:22:02.915 --> 00:22:07.673 Alright, from geometry you may remember that complementary angles 00:22:07.675 --> 00:22:10.680 are angles that have the sum of... Do you remember? 00:22:10.682 --> 00:22:16.430 Yes, 90 degrees or pi over 2. So let's remember that the word 00:22:16.432 --> 00:22:28.449 complementary means that two angles have the sum of 90 degrees or pi over 2. 00:22:28.451 --> 00:22:33.697 So in this right triangle, we have the sides labelled a, b, c is the hypotenuse. 00:22:33.699 --> 00:22:37.445 This is the angle theta, right angle. So the question is, 00:22:37.447 --> 00:22:41.710 what is this angle in terms of theta? 00:22:41.712 --> 00:22:45.212 Well, I gave it away a little bit when I talked about complementary angles 00:22:45.214 --> 00:22:49.961 because, if you remember, the sum of all three angles of a triangle is 180 degrees. 00:22:49.963 --> 00:22:54.723 So if it's a right triangle then that means that the sum of these two angles is, 00:22:54.725 --> 00:22:59.723 yes, 90 degrees which means these are complementary angles. 00:22:59.725 --> 00:23:05.230 So what that means is that I can label this angle right here, 00:23:05.232 --> 00:23:11.492 and I'm going to bring it out to here. I'm going to label that angle as what's left. 00:23:11.494 --> 00:23:17.490 The sum is 90, so I could say theta plus this angle is 90. So what's left 00:23:17.492 --> 00:23:23.754 for this one is 90 degrees minus what was used here, 00:23:23.756 --> 00:23:31.503 or I could write it as pi over 2 minus theta. 00:23:31.505 --> 00:23:37.004 Ok, now you have to get used to this notation so that when you see it you can spot it. 00:23:37.006 --> 00:23:43.023 So if this angle is theta, then the other acute angle in the right triangle could be 00:23:43.025 --> 00:23:48.526 written as 90 degrees minus theta, or it could be written as pi over 2 minus theta. 00:23:48.528 --> 00:23:54.531 Alright, now let's look at cofunctions of these complementary angles. 00:23:54.533 --> 00:24:00.040 Lets start with theta. With theta, let's just take the first one. 00:24:00.042 --> 00:24:06.292 So we'll say the sine of theta is... OK now remember the definition. 00:24:06.294 --> 00:24:10.797 The sine of theta is the opposite over the hypotenuse so that's 00:24:10.799 --> 00:24:16.057 the length of side b divided by the length of side c. 00:24:16.059 --> 00:24:22.556 Alright now, what is the cofunction for sine? Think about the word co-sine. 00:24:22.558 --> 00:24:30.320 The cofunction is the cosine. What is the value of the cosine? 00:24:30.322 --> 00:24:34.074 But we're not going to take the cosine of theta. We're going to look at 00:24:34.076 --> 00:24:39.831 the cofunction of the complementary angle, which is this one. 00:24:39.833 --> 00:24:44.835 Now, I can write it either way. I'm going to choose to write it as 00:24:44.837 --> 00:24:50.085 pi over 2 minus theta, but I could have put 90 degrees minus theta. 00:24:50.087 --> 00:24:56.096 And so what is the cosine of this angle? 00:24:56.098 --> 00:25:03.844 Alright, now think. Cosine is adjacent over hypotenuse. 00:25:03.846 --> 00:25:08.611 I'm using this angle, so this is the adjacent side. 00:25:08.613 --> 00:25:12.607 This is the opposite side. This is the adjacent side. 00:25:12.609 --> 00:25:18.617 So the cosine of this angle is b over c. 00:25:18.619 --> 00:25:22.626 OK, well as soon as I get to this point, what do you realize? 00:25:22.628 --> 00:25:38.377 You realize that cofunctions of complementary angles are equal. 00:25:38.379 --> 00:25:46.875 So what we have is this. We have that the sine of theta is equal to 00:25:46.877 --> 00:25:55.153 the cofunction of the complementary angle. 00:25:55.155 --> 00:25:58.657 Now, if we had all day we could go through all of the other trig functions, 00:25:58.659 --> 00:26:04.407 but I think by this process you can see that I could write a similar situation for, 00:26:04.409 --> 00:26:11.660 let's say, the tangent of theta, and I could write this as equal to 00:26:11.662 --> 00:26:19.657 the cotangent of the complementary angle, and I could continue along 00:26:19.659 --> 00:26:24.443 with secant and cosecant. So I could write that the secant of theta 00:26:24.445 --> 00:26:37.449 is the cosecant of the complementary angle. 00:26:37.451 --> 00:26:44.196 Alright, these and the rest of the relationships show you that 00:26:44.198 --> 00:26:49.400 cofunctions of complementary angles are equal. 00:26:50.959 --> 00:26:53.462 Let's take a look at an example of how 00:26:53.464 --> 00:26:57.713 the complementary angle theorem can be used to simplify an expression. 00:26:57.715 --> 00:27:00.473 OK, now you need to remember the fundamental identities, 00:27:00.475 --> 00:27:05.226 and I have some of the cofunctions of complementary angles 00:27:05.228 --> 00:27:08.975 identities still listed down here so we can refer to this. 00:27:08.977 --> 00:27:13.732 Ok, a says the cosine of 40 degrees divided by the sine of 50 degrees. 00:27:13.734 --> 00:27:18.741 OK, well it should jump out at you that the sum of 40 and 50 is 90, 00:27:18.743 --> 00:27:23.989 and so these are complementary angles. So what that means is we can rewrite 00:27:23.991 --> 00:27:30.250 this expression, let's go with the numerator, so the cosine of 40 degrees is equal to 00:27:30.252 --> 00:27:37.747 the cofunction of the complementary angle. And that's because if this angle 00:27:37.749 --> 00:27:44.752 is theta, then this angle is 90 degrees minus theta. And then we copy over the denominator, 00:27:44.754 --> 00:27:51.750 and immediately we see that what we have here is equal to 1. 00:27:51.752 --> 00:27:57.280 Alright, let's look at b. One plus the tangent squared of 5 degrees 00:27:57.282 --> 00:28:03.279 minus the cosecant squared of 85 degrees. OK, what hits you first? 00:28:03.281 --> 00:28:09.780 Well, maybe it's the fact that 5 degrees and 85 degrees are complementary angles. 00:28:09.782 --> 00:28:13.292 So we know somewhere we're going to use one of these. 00:28:13.294 --> 00:28:19.550 But we have tangent and cosecant, and none of this is going to be direct. 00:28:19.552 --> 00:28:23.800 So what we're going to have to do is find an identity, rewrite some of this, 00:28:23.882 --> 00:28:30.300 then use one of these complementary angle expressions. 00:28:30.302 --> 00:28:34.303 OK, there are a couple of ways you can go here. 00:28:34.305 --> 00:28:40.459 I would think that the most efficient approach would be to recognize right away 00:28:40.551 --> 00:28:45.578 that this expression can be rewritten as what? 00:28:45.580 --> 00:28:48.077 Now think about the Pythagorean identity. 00:28:48.079 --> 00:28:53.582 Tangent squared of an angle plus 1 is equal to what? 00:28:53.584 --> 00:29:01.090 Go through them. You should remember that it is secant squared of this angle 00:29:01.092 --> 00:29:12.591 which is 5 degrees. Now I have minus cosecant squared of 85 degrees. 00:29:12.593 --> 00:29:24.115 Well, what can I do now? How can I write say this one in terms of 5 degrees? 00:29:24.117 --> 00:29:28.869 Well this is cosecant. I could change either one of them actually. I just chose to do 00:29:28.871 --> 00:29:37.369 this one. I could rewrite this as the secant squared of 5 degrees, and then I know 00:29:37.371 --> 00:29:45.384 I've got it here because I've got the same two expressions. Subtract, and I get 0. 00:29:45.386 --> 00:29:49.888 Alright, let's take a look at another example. 00:29:51.146 --> 00:29:57.893 This time our angles are given in radian measure. c) We want to find 00:29:57.895 --> 00:30:04.899 the exact value of the sine of 5 pi over 12 divided by the cosine of pi over 12. 00:30:04.901 --> 00:30:10.914 Well, 5pi over 12 and pi over 12 are not special angles, so 00:30:10.916 --> 00:30:15.915 we're not going to be able to find the exact value exactly as the expression is written. 00:30:15.917 --> 00:30:21.913 So let's look at the two angles. I'm going to walk over here, and I'm going to 00:30:21.915 --> 00:30:34.162 examine 5 pi over 12 and pi over 12. What if I added the two angles? 00:30:34.164 --> 00:30:42.915 Well, I would get 6 pi over 12 which is pi over 2. 00:30:42.917 --> 00:30:47.950 So now what do we know about these two angles? 00:30:47.952 --> 00:30:53.699 They are complementary angles because the sum is pi over 2. 00:30:53.701 --> 00:30:58.708 What do we know about complementary angles? We know that cofunctions 00:30:58.710 --> 00:31:03.964 of complementary angles are equal. So let's look. Here are some of the 00:31:03.966 --> 00:31:12.465 cofunction identities. We can rewrite the sine of 5 pi over 12 as the 00:31:12.467 --> 00:31:32.463 cosine of pi over 2 minus 5 pi over 12. And then I'll just copy the denominator over, 00:31:32.465 --> 00:31:36.748 and because of what we did over here, we see that [the cosine of] pi over 2 00:31:36.750 --> 00:31:47.496 minus 5 pi over 12 is the cosine of pi over 12. The denominator is the cosine 00:31:47.498 --> 00:31:55.267 of pi over 12, and so the exact value of this expression is 1. 00:31:55.269 --> 00:32:03.015 d) We want to find the exact value of the tangent of pi over 6 minus 00:32:03.017 --> 00:32:10.017 the cotangent of pi over 3. So once again, let's look at the two angles. 00:32:10.019 --> 00:32:20.292 Pi over 6 plus pi over 3. Well we get a common denominator, so we write 00:32:20.294 --> 00:32:29.049 this is pi over 6, copying it over, we write this as 2 pi over 6. We add these. 00:32:29.051 --> 00:32:34.807 We get 3 pi over 6. We reduce, and we get pi over 2. 00:32:34.809 --> 00:32:41.062 So what have we found out? We found out here that pi over 6 and pi over 3 are 00:32:41.064 --> 00:32:46.560 complementary angles because their sum is pi over 2. 00:32:46.562 --> 00:32:57.564 So let's rewrite the tangent of pi over 6 using this second identity. It would be 00:32:57.566 --> 00:33:15.308 the cotangent of pi over 2 minus pi over 6, and then I copy this part over. 00:33:15.310 --> 00:33:23.312 Now this, according to what we saw here, can be rewritten as the cotangent of 00:33:23.314 --> 00:33:33.616 pi over 3. So what do we have here? 0. 00:33:33.618 --> 00:33:37.620 OK. Now let's move on to another example. 00:33:38.402 --> 00:33:45.624 Example 5: Given the cosecant of theta is 6, find the exact value of these 00:33:45.626 --> 00:33:50.629 four expressions. OK, now what you want to do here is use the most efficient method. 00:33:50.631 --> 00:33:56.381 We have the definitions. We have the reciprocal, Pythagorean, and quotient identities. 00:33:56.383 --> 00:34:00.391 And then we have the cofunction of complementary angles relationships. 00:34:00.393 --> 00:34:05.898 So we go for what's most efficient. Well, let's see. The cosecant and the sine 00:34:05.900 --> 00:34:13.147 are related. They are reciprocals. This 6 is a whole number. It has a denominator of 1. 00:34:13.149 --> 00:34:20.647 The reciprocal of that would be 1/6, so the sine of theta is 1/6. 00:34:20.649 --> 00:34:28.146 Alright, cotangent squared of theta. What do we know about cotangent squared 00:34:28.148 --> 00:34:33.676 that has a direct relationship to cosecant? OK, well we know this identity. 00:34:33.678 --> 00:34:42.175 We know that the cotangent squared of theta plus 1 is the cosecant squared of theta. 00:34:42.177 --> 00:34:49.187 We want to solve for cotangent squared, and that would be 00:34:49.189 --> 00:34:55.446 cosecant squared of theta minus 1. Alright, well we've just about got it because 00:34:55.448 --> 00:35:05.445 cosecant squared would be 6 squared, minus 1, so that would be 35. 00:35:05.447 --> 00:35:12.193 Alright, what about this one? Right away you should see that this angle, 00:35:12.195 --> 00:35:23.448 in these parentheses, this angle is the complement of the angle theta. 00:35:23.450 --> 00:35:32.966 So I can rewrite this expression. The cofunction of secant would be cosecant. 00:35:32.968 --> 00:35:37.958 And then the complement of this angle, this angle is 90 degrees minus theta, 00:35:37.960 --> 00:35:43.991 the complement of this angle is theta, and so what's the cosecant of theta? 00:35:43.993 --> 00:35:46.497 Given as 6. 00:35:46.499 --> 00:35:53.745 Alright, part d. Secant squared of theta. Hmm. OK, well it's certainly 00:35:53.747 --> 00:35:56.503 not going to have anything to do with cofunctions of complementary angles. 00:35:56.505 --> 00:36:02.508 And in terms of identities, there is no identity that directly relates 00:36:02.510 --> 00:36:08.016 secant or secant squared with cosecant. So I think at this point, 00:36:08.018 --> 00:36:11.768 the most efficient thing to do is to draw a triangle, label the sides, and then just 00:36:11.770 --> 00:36:18.027 read directly off of the triangle. So let's draw a triangle. 00:36:18.029 --> 00:36:22.783 It doesn't have to all be perfect here, but just enough to get the idea. 00:36:22.785 --> 00:36:29.532 This is the cosecant, so this is going to be the hypotenuse over the opposite, 00:36:29.534 --> 00:36:35.782 or you can read from the sine, opposite over hypotenuse. So this is 6, and this is 1. 00:36:35.784 --> 00:36:45.528 The missing side is 36 minus 1 is 35, and then we take the square root. 00:36:45.530 --> 00:36:51.808 Ok, so there are the three sides. Now, the secant is, well think about what that is. 00:36:51.810 --> 00:36:58.318 It's going to be the hypotenuse over the adjacent. The way I actually remember this 00:36:58.320 --> 00:37:00.819 is I think of the fact that it's the reciprocal of the cosine. 00:37:00.821 --> 00:37:04.323 Cosine is adjacent over hypotenuse, and then I take the reciprocal. 00:37:04.325 --> 00:37:09.568 OK, so what do I have here? And let's not forget that it is squared. 00:37:09.570 --> 00:37:15.330 So let's set up for that before we read, and we want hypotenuse, which is 6, 00:37:15.332 --> 00:37:32.853 over adjacent, which is square root of 35, and we square that so we get 36 over 35. 00:37:32.855 --> 00:37:36.857 So in summary, how do you find exact values? 00:37:36.859 --> 00:37:39.087 You use everything you know about this. 00:37:39.089 --> 00:37:43.089 You use the right triangle definitions of the trig functions when appropriate. 00:37:43.091 --> 00:37:47.366 You use the complementary angle theorem when appropriate, and you also use 00:37:47.368 --> 00:37:51.872 identities when that's the most efficient method. You need to study all of these 00:37:51.874 --> 00:37:58.100 relationships and master them until you are perfectly comfortable using them. 00:38:00.000 --> 00:38:07.385 Computing Trig Functions of Acute Angles Let's first talk about three acute angles. 00:38:07.387 --> 00:38:24.155 We call these special angles. Let's look at a 30 degree angle, which is pi over 6, 00:38:24.157 --> 00:38:34.915 a 45 degree angle, pi over 4, and then a 60 degree angle, pi over 3, 00:38:34.917 --> 00:38:39.665 and let's talk about finding the trig functions of these angles. 00:38:39.667 --> 00:38:52.414 Let's take 45 degrees first. Let's draw a triangle, and we will choose this triangle 00:38:52.416 --> 00:39:00.915 to be a right triangle, but we want it to be written so that it is an isosceles triangle 00:39:00.917 --> 00:39:10.451 where these two sides have the same length. Now it really doesn't matter 00:39:10.453 --> 00:39:14.204 what the length is of the two sides as long as they are the same. 00:39:14.206 --> 00:39:22.215 Let's try to make this look a little more like they're the same. 00:39:22.217 --> 00:39:28.472 So for convenience sake I will just say that the length of each side is 1. 00:39:28.474 --> 00:39:33.220 So what is the hypotenuse? Alright, let's see if we know how to do this. 00:39:33.222 --> 00:39:38.471 We're looking for c. So it's this squared, plus this side squared, 00:39:38.473 --> 00:39:41.233 add them together, and then take the square root. 00:39:41.235 --> 00:39:45.486 So it's going to be 1 squared plus 1 squared is 1 plus 1 which is 2, 00:39:45.488 --> 00:39:55.485 so this side is the square root of 2. Well, let's see. If this is an isosceles triangle, 00:39:55.487 --> 00:40:01.501 then that means that this angle right here is 45 degrees, and of course, 00:40:01.503 --> 00:40:06.006 that was by design because I want to study the trig functions of a 45 degree angle. 00:40:06.008 --> 00:40:12.264 So let's see what we have. Well we know that the sine, 00:40:12.266 --> 00:40:17.761 and of course this is pi over 4, and you want to think in radians as well as degrees. 00:40:17.763 --> 00:40:25.005 So we say that the sine of 45 degrees is the opposite over hypotenuse, 00:40:25.007 --> 00:40:30.778 and again there is no need to rationalize unless you are specifically told to do so. 00:40:30.780 --> 00:40:40.031 The cosine of 45 degrees is adjacent over hypotenuse, 00:40:40.033 --> 00:40:51.804 and the tangent of 45 degrees is 1 over 1 which is 1. 00:40:51.806 --> 00:40:57.299 These are the first three trig functions. You can get the next three by taking the 00:40:57.301 --> 00:41:03.810 reciprocal of each one of these. So we will just leave that to be done when we need it. 00:41:03.812 --> 00:41:10.064 Now let's look at a 30 degree angle and a 60 degree angle at the same time. 00:41:10.066 --> 00:41:17.074 So we have a 30 degree angle, pi over 6, and we're also going to look at a 00:41:17.076 --> 00:41:23.583 60 degree angle, and how are we going to do that? Well, we are going to draw 00:41:23.585 --> 00:41:33.835 an equilateral triangle or some approximation of an equilateral triangle. 00:41:33.835 --> 00:41:38.099 Now we have to be creative here. We know that if it's an equilateral triangle, 00:41:38.101 --> 00:41:43.604 all three angles are 60 degrees. I want to represent 60 and 30 and read from there. 00:41:43.606 --> 00:41:48.107 So what I'm going to do is, OK, I know this is 60 degrees. 00:41:48.109 --> 00:41:57.608 I'm going to drop a perpendicular which bisects this side. That means that this angle 00:41:57.610 --> 00:42:05.859 is 30 degrees, and what I'll do is I'll just say that the length of this side is 2, 00:42:05.861 --> 00:42:13.361 making the length of this half of the same length, 1. That's very convenient. 00:42:13.363 --> 00:42:16.107 Of course you could use any numbers, but whatever you do the ratio is that 00:42:16.109 --> 00:42:20.642 this will be twice that side. OK what is the length of this side? 00:42:20.644 --> 00:42:27.142 It's going to be 2 squared which is 4 minus 1 squared, so that's 4 minus 1 is 3. 00:42:27.144 --> 00:42:32.145 So the length of this side will be the square root of 3. 00:42:32.147 --> 00:42:37.143 Reading from the triangle. Alright, let's do 30 first, which means I'm going to read 00:42:37.145 --> 00:42:46.417 this angle. So this side is opposite. This side is adjacent. OK, so I'm going to do 00:42:46.419 --> 00:43:00.669 the sine of 30, and then I'm going to read the cosine of 30, and then the tangent of 30. 00:43:00.671 --> 00:43:11.190 Alright, the sine of 30 is opposite over hypotenuse. The cosine of 30 is 00:43:11.192 --> 00:43:21.201 adjacent over hypotenuse. And the tangent of 30 is, and you can either read it 00:43:21.203 --> 00:43:24.456 from the triangle, or we've done enough of this now that we know that 00:43:24.458 --> 00:43:29.210 it's going to 1 over the square root of 3 because we can use the identity, 00:43:29.212 --> 00:43:33.211 the reciprocal (quotient) identity that the tangent is the sine over the cosine. 00:43:33.213 --> 00:43:39.721 OK, we can get the other three using the reciprocal properties. 00:43:39.723 --> 00:43:46.224 Alright, now let's read the trig functions of the 60 degree angle. So we want 00:43:46.226 --> 00:43:55.977 the sine of 60 or pi over 3, we want the cosine of 60, and the tangent of 60. 00:43:55.979 --> 00:44:09.000 The sine is going to be opposite over hypotenuse, and then at this point 00:44:09.002 --> 00:44:14.749 you might start to realize that cofunctions of complementary angles are equal 00:44:14.751 --> 00:44:22.251 means that the cosine of 30 equals the sine of 60 and the sine of 30 degrees 00:44:22.253 --> 00:44:28.018 is the cosine of 60 degrees. We don't necessarily have to read all of this. 00:44:28.020 --> 00:44:34.772 And then the tangent of 60 is going to be the opposite over the adjacent 00:44:34.774 --> 00:44:41.780 which is the square root of 3, or the reciprocal of, well let's not. 00:44:41.782 --> 00:44:45.533 We're not going to have cotangent here. We're not going to write all those down. 00:44:45.535 --> 00:44:51.535 You can figure those out using the reciprocal. OK, so what do we have here? 00:44:51.537 --> 00:44:55.543 We have some things that would be very convenient to memorize. 00:44:55.545 --> 00:45:00.295 So what we want to do is we want to make this little table. 00:45:00.297 --> 00:45:03.291 And you want the table, and I'm just going to just write it in right here. 00:45:03.293 --> 00:45:10.794 You want the sine, the cosine, and the tangent, and this needs to go in here, 00:45:10.793 --> 00:45:19.047 of 30 degrees, 45 degrees, and 60 degrees. And this is something you want to write 00:45:19.049 --> 00:45:25.823 down over and over and over and over again until you are just totally 00:45:25.825 --> 00:45:34.581 comfortable with it. OK, so what do we have? We have the sine of 30. 00:45:34.583 --> 00:45:44.582 Now that's this batch right here. So we have the sine of 30 is 1/2, the cosine of 30 00:45:44.584 --> 00:45:52.331 is the square root of 3 over 2, and the tangent of 30 is 1 over the square root of 3. 00:45:52.333 --> 00:46:00.106 OK, next column would be these. The sine and cosine are the same. 00:46:00.108 --> 00:46:04.361 You see how easy this is to memorize. You just get comfortable with it. 00:46:04.363 --> 00:46:17.863 And then the tangent of 45 is 1, and then we have this last batch. 00:46:17.865 --> 00:46:25.885 And so we have the sine of 60, which of course is the cosine of 30, 00:46:25.887 --> 00:46:31.866 so these are easy. They're just switched, and then this one is the reciprocal. 00:46:31.868 --> 00:46:45.635 This right here is so important. You will use it over and over and over again. 00:46:45.637 --> 00:46:49.384 Let's take a look at some examples where we use this information. 00:46:50.402 --> 00:46:58.902 Example 1: Find the exact value of each expression. Notice that what we have here 00:46:58.904 --> 00:47:04.922 are all trig functions of acute angles, but not only that. All of these acute angles 00:47:04.924 --> 00:47:09.928 are special angles. Below I have the table that we generated by drawing 00:47:09.930 --> 00:47:16.685 the triangles for 45, 30, and 60 degrees. And finding the trig functions from there 00:47:16.687 --> 00:47:19.690 you can of course use the triangles or you can use this table. 00:47:19.692 --> 00:47:27.190 Alright, so let's see what we have. The sine of 45 degrees. OK, the sine of 45 degrees 00:47:27.192 --> 00:47:34.956 is 1 over the square root of 2. We want to multiply that by the cosine of 30 degrees. 00:47:34.956 --> 00:47:42.961 The cosine of 30 degrees is the square root of 3 over 2. 00:47:42.961 --> 00:47:50.900 Now what we're really doing here is practicing using this information. So we're going 00:47:50.902 --> 00:47:58.718 to multiply here, and we will get the square root of 3 over 2 square root of 2. 00:47:58.720 --> 00:48:02.970 Let me point out right here to you something about rationalizing. 00:48:02.972 --> 00:48:07.731 The sine and the cosine in particular say of 45 degrees 00:48:07.733 --> 00:48:09.237 are 1 over the square root of 2. 00:48:09.239 --> 00:48:14.495 You don't have to rationalize, but if you chose to do so, you would multiply 00:48:14.497 --> 00:48:18.995 this expression by the square root of 2 over the square root of 2 00:48:18.997 --> 00:48:23.751 which would then simplify and you'd get the square root of 2 over 2. 00:48:23.753 --> 00:48:28.503 What you want to do is recognize that if you ever see this expression 00:48:28.505 --> 00:48:34.002 it is equivalent to 1 over the square root of 2 and also to this one. 00:48:34.004 --> 00:48:38.771 The same thing for this. This is actually equivalent to the square root of 3 over 3, 00:48:38.773 --> 00:48:42.769 and so you may see that at some point too. Alright, let's go to b. 00:48:42.771 --> 00:48:49.516 The tangent of pi over 4. Ok well the tangent of pi over 4 is 1, 00:48:49.518 --> 00:48:57.285 and the sine of pi over 3 which is 60 degrees is the square root of 3 over 2. 00:48:57.287 --> 00:49:01.036 Now how can we simplify this? Well, we can get a common denominator, 00:49:01.038 --> 00:49:08.288 so we write 1 as 2 over 2 and then subtract the square root of 3 over 2. 00:49:08.290 --> 00:49:20.035 We can combine the terms in the numerator, over the denominator. 00:49:20.037 --> 00:49:26.067 Let's look at the third one. The tangent squared of pi over 6. 00:49:26.069 --> 00:49:34.317 Well the tangent of pi over 6 is 1 over the square root of 3, and what we want to do is 00:49:34.319 --> 00:49:41.817 from here we want to square this. Then we want to add sine squared of pi over 4. 00:49:41.817 --> 00:49:52.593 So the sine of pi over 4 is 1 over the square root of 2. Don't forget to square it. 00:49:52.595 --> 00:50:02.592 And what do we get? This would be 1 over 3 plus 1 over 2, common denominator, 00:50:02.594 --> 00:50:11.613 this would be 2/6, 3/6, exact value 5/6. 00:50:11.615 --> 00:50:16.863 Alright, so what you need to do is memorize the trig functions of the three 00:50:16.865 --> 00:50:21.868 special angles and be able to use them quickly and efficiently. 00:50:21.870 --> 00:50:26.377 But what happens if your acute angle is not one of those three special angles? 00:50:26.379 --> 00:50:31.130 Then you need to use a calculator. Let's take an example of that 00:50:31.132 --> 00:50:35.130 and work out some exercises using the calculator. 00:50:36.636 --> 00:50:43.637 Example 2: Find the value correct to two decimal places. Notice that these 00:50:43.639 --> 00:50:48.150 are all trig functions of acute angles, but none of these are special angles, and so 00:50:48.152 --> 00:50:51.650 we're not going to be able to read off the triangle or the table that we've memorized. 00:50:51.652 --> 00:50:56.402 Instead we're going to have to look up these values that are stored in our calculator. 00:50:56.404 --> 00:51:05.167 OK, the big thing here is units degrees, radians, degrees, radians. 00:51:05.169 --> 00:51:12.170 Your calculator must be in the correct mode in order for you to get the correct answer. 00:51:12.172 --> 00:51:19.170 Alright so this is the sine of 48 degrees. Make sure your calculator is in degrees. 00:51:19.172 --> 00:51:24.686 The default is usually degrees, but you should always check that. So I'm in degrees, 00:51:24.688 --> 00:51:29.184 and now I want the sine of 48 degrees. Now depending on what kind of calculator 00:51:29.186 --> 00:51:32.940 you have you may have to enter this, if you have sort of an old fashioned 00:51:32.941 --> 00:51:37.945 scientific calculator you may have to put the angle in first and then the trig function. 00:51:37.947 --> 00:51:40.946 If you get one that has a display, which is what I would recommend 00:51:40.948 --> 00:51:47.207 it's much easier to work with, you can depress the sine button and then you can 00:51:47.209 --> 00:51:52.964 put in 48. It's already in degrees, enter, and out comes the answer, 00:51:52.966 --> 00:51:59.466 and then I want to round correct to two decimal places. I get 0.743, 00:51:59.468 --> 00:52:10.463 and so when I round I get approximately decimal 74. Alright, what unit is this? 00:52:10.465 --> 00:52:16.712 Radians. So clear your calculator and put it in radian mode. 00:52:16.714 --> 00:52:22.242 OK, I'm bringing up radian mode, I am selecting it, and it now says radian mode. 00:52:22.244 --> 00:52:31.247 I want the cosine of pi over 12, so that's pi, and there's usually a key for pi, 00:52:31.249 --> 00:52:40.762 and then divided by 12, enter, and I get .9659 and so forth, rounded to two 00:52:40.764 --> 00:52:52.011 decimal places is going to be .96. OK, now let's see. This was 9659, so you know 00:52:52.013 --> 00:53:00.010 when I round this it should actually be .97. Be careful when you are rounding. 00:53:00.012 --> 00:53:10.511 OK, let's clear the calculator, and let's go back to degree mode. OK, so mode, degrees, 00:53:10.513 --> 00:53:18.264 enter. OK, I'm in degrees, but this is the secant of 17 degrees, and if you look at 00:53:18.266 --> 00:53:22.799 the calculator it has sine, cosine, and tangent. So how can I get the secant? 00:53:22.801 --> 00:53:26.551 Well, that's what we learned all those identities for. 00:53:26.553 --> 00:53:36.049 This is 1 over the cosine of 17 degrees. So let's enter that information. So it's 00:53:36.051 --> 00:53:48.553 1 divided by the cosine of 17, I'm in degrees, enter, and I get 1.045 blah blah, so 00:53:48.555 --> 00:53:52.832 this is going to have to be approximate because I'm rounding, and I'm going to get 00:53:52.834 --> 00:54:00.800 1 point 0 and I have 4 5, so I'm going to round to 1.05. 00:54:00.802 --> 00:54:04.340 OK, what about this one? Mode? 00:54:04.342 --> 00:54:13.073 Correct. So clear the calculator, go back to radian mode. Radian mode, enter. 00:54:13.075 --> 00:54:18.352 The cosecant is 1 over, and I have to rewrite this in terms of either 00:54:18.354 --> 00:54:22.106 sine, cosine or tangent. And what is it? 00:54:22.108 --> 00:54:31.102 Mmhm. Sine of pi over 5. So I'm in radians and I'm ready. It's 1 divided by 00:54:31.104 --> 00:54:41.876 the sine of pi divided by 5 and that is equal to 1.701, so that is going to be 00:54:41.878 --> 00:54:50.380 approximately 1.70 correct to two decimal places. And so this is the process you use 00:54:50.382 --> 00:54:55.140 to find the trig functions of acute angles when they are not special angles. 00:54:55.142 --> 00:55:00.645 You use your calculator. If you have a special angle, either 30, 60, or 45 degrees, 00:55:00.647 --> 00:55:05.700 you should know that information by heart. Practice it until you do.