 Welcome back to our lecture series math 1060 Trigonometry for students at Southern Utah University. As usual, I'll be a professor today, Dr. Andrew Misildine. In lecture 15, we're going to begin chapter 6, which is a very important chapter about trigonometric identities. Now trigonometric identities, really what an identity is, it's just an equation for which any evaluation of the variable will always make the statement true. An identity is exactly that. It's saying something like, if you take a plus b squared, this equals a squared plus 2ab plus b squared. It doesn't matter what you choose for the real numbers a or b, the left-hand side and the right-hand side will always equal each other. That's what we mean by here in identity. Identities are important because they help us recognize when two mathematical expressions are equal to each other. Trigonometric identities, as the name suggests, focus on identities which we can guarantee that trigonometric expressions are fact equal to each other. What I wanted to do in this video is review trigonometric identities we've run across previously in this lecture series, and we're going to label these the fundamental or basic identities. It turns out that every other trigonometric identity that we learn will essentially be derived from these basic identities with perhaps a few exceptions. What are these fundamental identities? These are identities that every student of trigonometry really does need to know. Now, to make it a little bit easier to remember them, I'm going to put them in families based upon certain types. The first family of fundamental identities is going to be called the reciprocal of identities, reciprocal family. The name comes from the fact that these are reciprocal statements. We've learned previously that cosecant is the reciprocal of sign. That is to say cosecant theta equals one over sign theta. Likewise, secant is the reciprocal of cosine. More specifically, secant theta is equal to one over cosine theta. And lastly, cotangent is the reciprocal of tangent. That is, cotangent theta equals one over tangent theta. This gives us the reciprocal family. And so because of these reciprocal families right here, we often put emphasis on sine, cosine, and tangent. Like when we were doing right triangle trigonometry earlier in this lecture series, we focused a lot on so catoa, sine, cosine, tangent. We kind of ignored cosecant, secant, and cotangent. Why is that? Well, because of the reciprocals of sine, cosine, and tangent, any question that could be answered with cosecant, we probably could have used sine. Or any problem we could have solved with secant, we probably could have used cosine as well. And so there's a lot of preference on sine and cosine. Sine, cosine, tangent because of these reciprocal relationships. The next family is the ratio family. The ratio family tells us that tangent can be written as a ratio of sine and cosine. More specifically, tangent theta is equal to sine theta over cosine theta. So the reciprocal identities put preference on sine, cosine, tangent because cosecant, secant, and cotangent can be translated into the language of sine, cosine, and tangent. But the ratio identity tells us that we can write every tangent using just sine and cosine. Therefore, there's a lot of preference that's placed on sine and cosine. In some regard, they're the fundamental units of trigonometry. Every trigonometric expression can be expressed just using sine and cosines. We'll actually do some practice of that later on in this video. Now, when you look at this table here, you have this column identified as the basic identities. But then you have this other column called the derivations, the children of these. What does that even mean? Well, it turns out that there is a litany of trigonometric identities. There is a lot of symmetry, so to speak, when it comes to trigonometric functions. We saw a lot of that when we were graphing trigonometric graphs, of course. There are different ways to represent the same graph using two seemingly different trigonometric functions. There's a lot of symmetry when it comes to trigonometric functions, which means there's a lot of trigonometric identities. They would be impractical and somewhat cruel to expect a student to memorize every possible trigonometric identity under the sun. I don't know every identity, right? But with that said, if we have a certain small amount of critical mass of trigonometric identities, we then could derive all the other trigonometric identities from them. So for example, if you know that cosecant is the reciprocal of sine, and you also know that sine is the reciprocal of cosecant, more specifically sine theta equals one over cosecant theta. How do I know that? Well, you know that cosecant is the reciprocal of sine, and you know a very fundamental algebraic principle. Reciprocals is a two-way street. That is, reciprocity is a dual relationship. If I am your reciprocal, then you are my reciprocal. And so you can reciprocate this relationship into that sine is the reciprocal of cosecant. Similarly, since we know secant is the reciprocal of cosine, we also know that cosine is the reciprocal of secant. And since tangent is the reciprocal of, excuse me, since cotangent is the reciprocal of tangent, we also know that tangent is the reciprocal of cotangent. So we can combine these, we can use these identities to create new identities, and that's why we might call them their children. They come from these fundamental identities that we know here. When it comes to the ratio identity, well, since tangent can be recognized as a ratio of sine and cosine, and since cotangent is the reciprocal of tangent, if we combine these identities together, you know, their powers combine, I am Captain Planet, not exactly that, but if you get one over tangent, that's cotangent, but tangent is sine over cosine. So you get one over sine over cosine, and then applying algebraic properties of taking the reciprocal of a fraction, then you just flip that fraction upside down, you get cosine over sine. So cotangent theta is equal to cosine theta over sine theta. You know, like sometimes I'm labeling, I'm reading this word theta out loud, write the variable, it is important. When we speak, we can get away with things like cotangent equals cosine over sine, but when you write it down, you really do need to write the angles, especially as we in the future, the angle can be very ambiguous. If we're not explicit about it, this can lead to problems. So writing things like cotangent equals cosine over sine, again, that's how we say it in language, but that's not how we should write it mathematically. This is a very egregious typographical error. I mean, when you write something like this, there's a mathematical fairy that dies somewhere. So don't do that. Always write down the angle when you write a trigonometric expression. The third family, which is actually the biggest, even though in terms of the fundamental list, it's really the shortest, is the Pythagorean family, the Pythagorean identity. The Pythagorean identity gets its name because the fundamental identity here is cosine squared plus sine squared equals one. We saw this when we were studying the unit circle and we saw that sine and cosine have this Pythagorean relationship when we start considering right triangles that are embedded inside of the unit circle. Cosine squared plus sine squared equals one. This is just the Pythagorean equation, A squared plus B squared equals C squared applied to the unit circle. Now from this identity, we can derive so, so many new identities and all of them which are very, very useful in the right context, all right? And I like to use the karate kid as an analogy here because when Daniel Son was first learning karate from Mr. Miyagi, Mr. Miyagi starts off giving all these chores like paint the fence, sand the deck, wax on, wax off, all these different things, which they seem like annoying chores that some old man was just exploiting a teenager to do all these housework with. But it turns out it wasn't. He was practicing certain muscle skills, which at a context looks like chores for the old guy, but in the right defensive context, we learned we're actually very important defensive karate techniques, which if you don't want to get your butt kicked by the Cobra Kai all the time, then defense is actually the very first thing you should learn with karate. And so that's the same thing going on right here that we oftentimes don't see the settings where we need these trigonometric identities, but I promise you those settings are there and the context will come if you're patient with your sensei right now, which of course is me. I'm a sensei missile down right now. Anyways, if you take the Pythagorean identity, if you want to solve for sine squared, it's pretty easy to do, just subtract cosine squared from both sides. You get sine squared is equal to one minus cosine squared. It's a very simple derivation, but why might it be important? Well, what this identity tells us is that if you have a difference of squares, one minus cosine squared, this can actually be turned into a perfect square, sine squared. We'll actually see very shortly in another video why that is actually a very useful skill. If you take the square root of both sides, you get sine theta equals plus or minus the square root of one minus cosine squared, where the plus or minus depends which quadrant we're in, but we can write sine as some kind of a combination of cosine. So if you didn't want to sign, you could write just with cosine. So we mentioned earlier how every trigonometric expression can be written with sine, cosine and tangent, which you can also kick out tangent just to get sine or cosine. But it turns out you can also kick out sine and just write every expression in terms of cosine. Cosine's the only trigonometric identity, excuse me, trigonometric function you need. Now, don't get me wrong, doing this can get really complicated. The square root is maybe not worth the trade-off. But again, there might be some setting where only having cosine is a good idea. But you can also go the other way around. Maybe you want to move sine squared to the other side and you end up with cosine squared equals one minus sine squared. Or take the square root, you get cosine equals plus or minus, depending on the quadrant, the square root of one minus sine squared. So you don't even need a cosine because every cosine could be expressed with sine. So every trigonometric expression could be written using sines and cosines or just sine or just cosine. These next two children of the Pythagorean identity, and you'll notice this identity has lots of children. These are just the ones I'm listing on the screen. These have so many, this Pythagorean identity is the most fundamental of all identities, the most important, has the most children that some people refer to this as the mother of all trig identities or some people call it the mom identity because she has so many children. Another very important derivation of this is the following, one plus tangent squared equals secant squared. And to derive this one, there's a nice little trick here. You're just going to divide everything by cosine squared. If you divide everything by cosine squared, assuming I can fit it in there, what do you get? Cosine squared over cosine squared is a one. Sine squared over cosine squared, we'll sine over cosine is tangent, remember? And so sine squared over cosine squared becomes a tangent squared. And one over cosine, remember one over cosine is secant, so you end up with a secant squared right there. So if you take the mother identity and divide her by cosine squared, you get one plus tangent squared equals secant squared, which this is also a very useful identity where you can take a sum of squares and turn it into a perfect square, one plus tangent squared equals secant squared. Similarly, if you divide the mom identity by sine squared, you can get something similar to that. Cosine squared over sine squared, well, cosine over sine is cotangent, so that becomes a cotangent squared. Sine squared over sine squared, that gives you a one that you see right here. And then lastly, one over sine, that's not on the screen anymore, but that was cosecant, so you get cosecant squared. So all of these are children of the mother identity, such as cotangent squared plus one equals cosecant squared. You don't necessarily have to memorize this identity. If you remember this identity, you're like, oh, I can make, what would the identity involving tangent squared and secant squared? Well, you took the mother identity and divided it by cosine squared, boom, you can re-derive it, and there you go. So a few more identities we wanna mention here. There's the co-function theorem gives us the complementary identities. So sine and cosine are compliments to each other. So sine theta is equal to cosine of pi halves minus theta, which notice theta and pi halves minus theta, these are complementary angles. So if you take sine and switch to its complementary, excuse me, if you take an angle and switch to its complementary angle, then sine and cosine are equal for those. This would say things like, sine of 30 degrees is equal to cosine of 60 degrees. Likewise, cosine and sine are complementary functions. Cosine of say 70 degrees is equal to sine of 20 degrees, things like that. And then also we can derive from them very quickly that cosecant and secant are complementary functions, tangent and cotangent are complementary functions. Remember, that's what the co is for. Co is short for complementary. Cosine is the complementary sine. Cosecant is the complementary secant. You get the very basic pattern right there. All right, so this gives us another way how you could write everything in terms of sine if you didn't want cosine. But this gets a little bit messy because it changes the angle. But also this explains a reason why before, why it's important to always write the angle. Because if you drop the angle here, you get sine is equal to cosine, which is false, you know, in general, right? It's like, are you saying that sine theta is equal to cosine theta? Well, that only happens when you have a self-complimentary angle like 45 degrees or a pi force if you prefer in radians. The last of the fundamental identities is the symmetry identities, which tells us that sine and cosine are symmetric. Sine is an odd function. Sine of negative theta is equal to negative sine theta. And cosine is an even function. Cosine of negative theta is equal to cosine theta. Using the reciprocal and ratio identities, we also get that cosecant is odd, secant is even. Tangent is an odd function. This is an easy one to see right here. Because the idea is tangent of negative theta is equal to sine of negative theta over cosine of negative theta. Since sine is odd, it spits out the negative sine. Since cosine is even, it just gobbles up the negative sine, it disappears. In which case now you just have a negative in front of tangent using the ratio identity again, which was exactly this friend right here. A similar statement can be said for cotangent. It's an odd function right there. So these are our fundamental identities, which I'm gonna zoom out just for a second so you can take one last look at them before we look at some other things, of course. Can I get them all in one view right there? There you go. So we have the reciprocal, ratio, Pythagorean, complementary and symmetric identities. These ones are ones you need to know. You don't necessarily need to know all the children because the children, as we illustrated in some of the cases, can be easily derived from these fundamental identities. Let's put them into practice now. So some things we know about these fundamental identities, if ever you have a trigonometric expression that involves a fraction or ratio, it turns out that all of those can be turned into products. You never have to have fractions when it comes to trigonometry or you can make fractions if you need to. So for example, how can we rewrite secant theta divided by tangent theta as a product? Well, whenever you have a trigonometric function in the denominator like this, you can move it to the numerator by choosing its reciprocal. So secant theta over tangent theta, this is the same thing as secant theta times cotangent theta. So if you prefer a product over a fraction, the ratio and particularly the reciprocal identities come into play here. So every fraction can be rewritten as a product. And like I also mentioned, every trigonometric function can be written using just signs and cosines. So if you take the product secant times tangent, how do you write that using signs and cosines? Well, secant theta times tangent theta. Well, secant theta can be rewritten as one over cosine theta using the reciprocal identities. And tangent using the ratio identity is the same thing as sine theta over cosine theta, like so, for which you get one times sine, which is sine, and you get cosine times cosine, which gives you a cosine squared. And so sine over cosine squared is the same thing as secant times tangent. And maybe this is advantageous. Now you do have a fraction here, and so this is sort of the trade-off. You can't get everything. You can't have your cake and eat it too, right? If you don't want a fraction, you might need to use all the trigonometric functions. But if you only want signs and cosines, you might have to use a fraction. It depends on the setting what you're gonna do. And that's the thing is this gives us a choice. We have the option to do it, and we would choose that option if the option is somehow advantageous to us. So let's take the, just take the trigonometric function tangent. How do we write this using only signs and cosines? Well, that's a pretty easy one because by the ratio identity, tangent theta is the same thing as sine theta over cosine theta. Well, could we redo this using only sine? Only sine? If you're only using sine, then will the sine of the numerator is perfectly fine, but the cosine of the denominator, we could write this as plus or minus the square root of one minus sine squared. Be aware that that expression, the denominator does not simplify because you can't distribute an exponent like a square root across a sum or difference. I'd probably put the plus or minus out in front so you get tangent is equal to plus or minus sine theta over the square root of one minus sine theta, sine squared, excuse me, for which you plus or minus depending on the quadrant, tangent will be positive in the first and third quadrant. It's negative in the second and fourth quadrant. So it depends on the angle right there. So we can rewrite this expression using just signs or cosines are in this case just a sign. Let's do one more example of rewriting trigonometric expressions using these fundamental identities. Let's rewrite the expression secant theta divided by tangent theta in terms of signs and codesides. And then we're gonna simplify it for which you might recognize this one before if it's like I feel deja vu happening right now. It's gonna secant tangent. Didn't we do it a moment ago? We did, right? So secant divided by tangent. It doesn't tell you you have to move it into a product but if you're trying to turn everything into signs and cosines because tangent itself is a ratio of sine and cosine, it's in the denominator. If you just immediately turn it into fractions right now you'd end up with like one over cosine theta over sine theta over cosine theta, right? That's perfectly fine. You have a fraction divided by fractions. You get one over cosine theta times cosine theta over sine theta. That's perfectly acceptable. But I know for many, for many trigonometric students we're still struggling with basic techniques of fractions. So an alternative path that one could have chosen is instead of keeping as a fraction we could have turned into a product first that we did before secant theta times cotangent theta. Then you're gonna get one over cosine like we saw before and then cotangent is cosine theta over sine theta. So you get the exact same thing that we did before but it avoids the nested fractions, fractions to the side of fractions. So you can see that this now provided a context where a product might have been more preferable than a fraction. So we switch it to a fraction then we switch it to signs and cosines that might simplify the algebra and play here. Whichever path you chose, it doesn't matter. Our goal is to simplify. You'll notice that there's a cosine theta on top and bottom. So you can cancel that out and you can end up with a one over sine theta. So if you're only trying to express it in terms of signs and cosines, you get one over sine. But we can even do better than that. One over sine is equal to just cosecant theta. And so now what we've observed is that secant theta over tangent theta is equal to cosecant theta. And so we've now proven a new trigonometric identity that secant divided by tangent is the same thing as cosecant. Or the other way, cosecant can be factored as secant divided by tangent. And so this is now starting to illustrate exactly why we use these fundamental identities and how one can start proving new trigonometric identities which will be a great goal for us later on in chapter six.