WEBVTT 00:00:07.500 --> 00:00:14.911 Let's take a look at trigonometric identities. An identity is a type of equation where f of x equals g of x 00:00:14.913 --> 00:00:21.867 for every value of x for which both functions are defined. Look at these examples over here on the left. 00:00:21.869 --> 00:00:28.621 X plus 1 quantity squared equals x squared plus 2x plus 1. It doesn't matter what value of x you plug into it. 00:00:28.623 --> 00:00:34.865 The left side will always equal the right side. The same is true for this bottom equation. This is one you should be 00:00:34.867 --> 00:00:40.870 familiar with. It's one of the Pythagorean identities. Cosine squared x plus sine squared x equals 1 for every value 00:00:40.872 --> 00:00:48.649 of x. If you plug in say x equals pi over 6, cosine squared pi over 6 plus sine squared pi over 6 will equal 1. 00:00:48.651 --> 00:00:55.236 Again, any value you plug in here for x, if you take cosine squared x plus sine squared x, you will always get 1. 00:00:55.238 --> 00:01:01.079 It's true for all values of x. In contrast, look at these two examples. They're not true for all values of x. 00:01:01.081 --> 00:01:07.830 We can easily see that by just say plugging in x equals 0. We can see on the left that 3 does not equal 6. The 00:01:07.832 --> 00:01:16.515 same thing on the bottom equation, cosine of 0 squared is not equal to 1/2. These are not true all the time. 00:01:16.517 --> 00:01:22.510 An identity is true for all values of x in the domain. Now there are two important things that you need to be 00:01:22.512 --> 00:01:29.606 able to do to be successful in establishing identities. One of them is that you need to know the basic trig identities. 00:01:29.608 --> 00:01:35.353 For example, you need to know the quotient identities, the reciprocal identities, and the Pythagorean identities, 00:01:35.357 --> 00:01:41.605 and the other thing that you need to be able to do is to manipulate expressions using algebra. For example, 00:01:41.607 --> 00:01:46.603 you need to be able to factor, you need to be able to reduce, you need to be able to get a common denominator. 00:01:46.605 --> 00:01:49.605 Let's take a look at an example. 00:01:51.600 --> 00:01:57.352 Here we want to multiply and simplify this expression. Now you need to be careful because there's a negative sign 00:01:57.354 --> 00:02:03.100 in front of this product. So you need to make sure that you multiply, and then you take the opposite of that product. 00:02:03.102 --> 00:02:09.828 So let's begin. First we're going to multiply cosine theta plus sine theta times cosine theta plus sine theta. 00:02:09.830 --> 00:02:14.532 I'm going to put a bracket here in front of my product. So what do we get when we multiply? We're going to get 00:02:14.534 --> 00:02:27.354 cosine squared theta plus 2 cosine theta sine theta plus sine squared theta, all over 00:02:27.356 --> 00:02:36.850 sine theta cosine theta. Now look inside the bracket. What do you see? I see a cosine squared theta plus 00:02:36.852 --> 00:02:41.613 a sine squared theta. Again, knowing those basic identities is very important, or you won't be able to 00:02:41.615 --> 00:03:00.864 recognize that. Cosine squared theta plus sine squared theta is 1, so I'll replace that. Now I'm ready 00:03:00.866 --> 00:03:08.889 to subtract what's inside the bracket, and let's see what we get. The 1's are going to subtract out, 1 minus 1. 00:03:08.891 --> 00:03:14.649 You're going to have the opposite. You're going to have minus 2 cosine theta sine theta 00:03:14.651 --> 00:03:21.650 in the numerator over sine theta cosine theta in the denominator, and now you recognize that we've 00:03:21.652 --> 00:03:26.649 got common factors in the numerator and denominator. We can reduce by sine theta and cosine theta 00:03:26.651 --> 00:03:34.500 and end up with just a negative 2. Alright, let's look at another example. 00:03:37.100 --> 00:03:43.111 Here we want to multiply and simplify this expression. Well if we didn't have to multiply to simplify, of course, 00:03:43.113 --> 00:03:49.600 this is just equal to 1, so we could get this. But you're going to see after we multiply that we get a nice expression. 00:03:49.602 --> 00:03:53.880 Let's go ahead and start. So we want to multiply the numerator. We're actually not going to multiply it out. 00:03:53.882 --> 00:04:01.401 We're just going to leave it this product. Now what is cosine theta plus 1 times cosine theta minus 1? 00:04:01.403 --> 00:04:10.651 Of course, that's cosine squared theta minus 1. This should sort of be recognizable to you because you 00:04:10.653 --> 00:04:23.586 know one of the Pythagorean identities is cosine squared theta plus sine squared theta equals 1. 00:04:23.588 --> 00:04:28.335 Well, I want to know what's cosine squared theta minus 1. So of course if I subtract 1 from each side 00:04:28.337 --> 00:04:34.081 and subtract [sine squared theta], I'm going to get that cosine squared theta minus 1 equals negative 00:04:34.083 --> 00:04:47.583 sine squared theta. So I can replace the denominator with negative sine squared theta. Now let's take a 00:04:47.585 --> 00:04:51.863 look. We notice that we have a factor of sine theta in the numerator and a sine theta in the denominator. 00:04:51.865 --> 00:05:02.873 We can reduce, and so we're left with cosine theta minus 1 in the numerator over negative sine theta 00:05:02.875 --> 00:05:09.624 in the denominator. So again, many times we'll multiply by some expression of 1 to try to establish the 00:05:09.626 --> 00:05:15.870 identity and get the other side. Alright, let's take a look at establishing identities. 00:05:17.988 --> 00:05:24.240 Here we want to establish the identity tangent theta plus cotangent theta times cosine theta equals 00:05:24.242 --> 00:05:28.986 cosecant theta. Well when we establish an identity, it's important that we work with only one side at a 00:05:28.988 --> 00:05:34.657 time. In this example, I'm going to work with the left-hand side, so I'm going to rewrite that side. 00:05:34.659 --> 00:05:42.658 Tangent theta plus cotangent theta times cosine theta. Now in this example what I'm going to do, 00:05:42.660 --> 00:05:48.659 which is done many times, is I'm going to change everything to sines and cosines. Remember that tangent theta is 00:05:48.661 --> 00:06:05.689 sine theta over cosine theta. Cotangent theta, using the quotient identity, is cosine theta over sine theta. 00:06:05.691 --> 00:06:10.188 Now inside the parentheses, I want to add those terms together, so I need to get a common denominator, 00:06:10.190 --> 00:06:25.452 which of course is cosine theta sine theta, which means that I'm multiplying numerator and denominator 00:06:25.454 --> 00:06:33.465 here by a sine theta, so I get sine squared theta for the first term. Here I'm multiplying by a cosine theta, 00:06:33.467 --> 00:06:41.725 so I get cosine squared theta. Look at the numerator, sine squared theta plus cosine squared theta. 00:06:41.727 --> 00:06:56.987 That should be familiar to you. That equals 1. Now when we multiply we see that we can reduce by a 00:06:56.989 --> 00:07:05.493 factor of cosine theta. We get 1 over sine theta, and of course, 1 over sine theta, the reciprocal identity, 00:07:05.495 --> 00:07:11.000 the sine and cosecant are reciprocals of each other, so this equals the cosecant which, of course, is the 00:07:11.002 --> 00:07:17.760 right hand side. So your goal is always to take one side and do algebra to it to get to the other side. 00:07:17.762 --> 00:07:19.752 Let's look at another example. 00:07:22.700 --> 00:07:28.263 Establish the identity 1 minus secant theta over tangent theta minus tangent theta over 1 minus secant 00:07:28.265 --> 00:07:35.028 theta equals 2 cotangent theta. What I'm going to do is work on the left hand side, and we want to combine 00:07:35.030 --> 00:07:39.532 the terms. We want to subtract, so we have to combine the terms. We need to get a common denominator, 00:07:39.534 --> 00:07:49.528 which would be tangent theta times 1 minus secant theta, which means that I'm multiplying numerator 00:07:49.530 --> 00:07:54.779 and denominator here by 1 minus secant theta. So I have 1 minus secant theta times another 1 minus 00:07:54.781 --> 00:08:01.776 secant theta, so I get 1 minus secant theta quantity squared. On the second term I'm multiplying by 00:08:01.778 --> 00:08:10.810 tangent theta over tangent theta, so I get minus tangent squared theta. Now we want to square this term. 00:08:10.812 --> 00:08:14.315 Remember when you square that term you're going to end up with three terms. We're going to square 00:08:14.317 --> 00:08:23.816 1 minus secant theta, and we get 1 minus 2 secant theta plus secant squared theta minus tangent squared 00:08:23.818 --> 00:08:36.587 theta all over tangent theta times 1 minus secant theta. Now look at the numerator. Remember one of 00:08:36.589 --> 00:08:46.338 the Pythagorean identities says that 1 plus tangent squared theta equals secant squared theta. So what 00:08:46.340 --> 00:08:52.355 you see is secant squared theta minus tangent squared theta, so if I subtract tangent squared theta from 00:08:52.357 --> 00:09:05.367 both sides I'm going to get 1. So I can replace all of this with a 1. So let's see what we get. We get 00:09:05.369 --> 00:09:19.131 1 minus 2 secant theta plus 1 over tangent theta [times] 1 minus secant theta, and of course 00:09:19.133 --> 00:09:46.631 1 plus 1 equals 2. Now here in the numerator we've got a common factor of 2. We can factor that out. 00:09:46.633 --> 00:09:51.625 Now we've got a common factor, numerator and denominator, of 1 minus secant theta. We can reduce 00:09:51.627 --> 00:09:59.381 by 1 minus secant theta, and we end up with 2 over tangent theta, and remember the reciprocal of tangent 00:09:59.383 --> 00:10:08.209 is cotangent. So this is the same thing as 2 cotangent theta, which of course is the right-hand side. So again, 00:10:08.211 --> 00:10:13.929 what you can see, it's important to be able to square terms. It's important to be able to recognize identities. 00:10:13.929 --> 00:10:21.175 So the algebra and the trig are very important skills in establishing these identities. Let's look at another example. 00:10:23.390 --> 00:10:29.650 In this problem we want to establish the identity 1 minus cosine theta over 1 plus cosine theta equals 00:10:29.652 --> 00:10:35.326 parentheses cosecant theta minus cotangent theta, quantity squared. Now remember when establishing 00:10:35.328 --> 00:10:41.827 identities, the rule of thumb is to pick the more complicated side. Well in this case, it's pretty close so I'm 00:10:41.829 --> 00:10:46.582 going to choose the right-hand side. But you could start with the left hand side, and what you would want 00:10:46.584 --> 00:10:52.582 to do is to multiply by the conjugate of the denominator, which is 1 minus cosine theta. So you'd multiply 00:10:52.584 --> 00:10:58.593 numerator and denominator by 1 minus cosine theta. I'm going to start with the right-hand side, 00:10:58.595 --> 00:11:05.101 cosecant theta minus cotangent theta, the quantity squared. Now when you square that you get 00:11:05.103 --> 00:11:16.851 cosecant squared theta minus 2 cosecant theta cotangent theta plus cotangent squared theta. 00:11:16.853 --> 00:11:23.120 Now what I'm going to do at this point is I'm going to change everything to sines and cosines. Cosecant 00:11:23.122 --> 00:11:32.368 squared is 1 over sine squared. Cosecant again is 1 over sine, so I have 2 over sine theta. Cotangent theta 00:11:32.370 --> 00:11:43.638 is cosine theta over sine theta, and cotangent squared theta is cosine squared theta over sine squared theta. 00:11:43.640 --> 00:11:48.895 Now let's see what we have. Notice that each of the denominators is sine squared theta, which means we 00:11:48.897 --> 00:11:55.653 can combine the terms, combine the numerators, to get everything over sine squared theta. So we have 00:11:55.655 --> 00:12:06.900 1 minus 2 cosine theta plus cosine squared theta. Now look at the numerator. You should notice that 00:12:06.902 --> 00:12:17.420 that factors, and it factors into 1 minus cosine theta, quantity squared. Now looking at the other side, 00:12:17.422 --> 00:12:22.173 which is always my guide, I see cosines. Remember you can always write a sine squared theta in terms 00:12:22.175 --> 00:12:29.433 of cosine squared because sine squared is 1 minus cosine squared. So now we're almost there. The only 00:12:29.435 --> 00:12:38.629 thing we need to do is to factor the denominator. 1 minus cosine squared theta factors into 1 minus 00:12:38.631 --> 00:12:47.956 cosine theta times 1 plus cosine theta, and now we can see that we have a common factor of 1 minus 00:12:47.958 --> 00:12:53.954 cosine theta. We can reduce by a factor of 1 minus cosine theta, and we're left with 1 minus cosine theta 00:12:53.956 --> 00:13:00.962 in the numerator over 1 plus cosine theta in the denominator, which of course you can see is the 00:13:00.964 --> 00:13:06.722 left-hand side. That's always your goal, to get to the other side. Alright, so remember that sometimes you 00:13:06.724 --> 00:13:13.222 want to multiply by the conjugate of the denominator, which in this case would be 1 minus cosine theta 00:13:13.224 --> 00:13:17.481 over 1 minus cosine theta. The problem could have been done that way, and you would have basically 00:13:17.483 --> 00:13:24.482 just reversed these steps to get to the right-hand side. Alright, let's look at another example. 00:13:26.906 --> 00:13:32.158 In this problem we want to establish the identity sine theta minus secant theta over sine theta secant theta 00:13:32.160 --> 00:13:37.918 equals cosine theta minus cosecant theta. In this problem I'm going to start with the left-hand side and 00:13:37.920 --> 00:13:48.180 what I'm going to do, first let me just write this side. Many times what we want to do is combine the terms. 00:13:48.182 --> 00:13:52.930 In this problem what I want to do is separate the terms. So what I'm going to do is separate this into two 00:13:52.932 --> 00:14:03.197 terms. This is the same thing as sine theta over sine theta secant theta minus secant theta over sine theta 00:14:03.199 --> 00:14:08.448 secant theta because remember one of the common mistakes a student might make is to think that 00:14:08.450 --> 00:14:14.952 there's a common factor. But there's not a common factor to each of these three terms. Now when I separate it 00:14:14.954 --> 00:14:22.714 into these two terms, I notice that in this term I can reduce by a factor of sine theta to get 1 over secant theta, 00:14:22.716 --> 00:14:29.252 and here I can reduce by a factor of secant theta and get 1 over sine theta, and then knowing the 00:14:29.254 --> 00:14:37.214 reciprocal identities, 1 over secant is cosine theta, and 1 over sine is cosecant theta, which of course you 00:14:37.216 --> 00:14:43.712 can see is the right-hand side. Now as you can see from these examples, it's very important that you 00:14:43.714 --> 00:14:50.717 know the basic trig identities very well so that when you see an expression you know what to replace it 00:14:50.719 --> 00:14:56.247 with, like sine squared theta plus cosine squared theta equals 1. And you need to go through these 00:14:56.249 --> 00:15:02.200 examples, practice them until you know how to do them very well without any assistance.