PC 11.2 Negative and Pythagorean Identities

So tricky, but not true myth number one is busting. All right, so let's take a look here the very first thing we're gonna talk about today are identities for negatives, all right? Well, sometimes get some negative angles in here. So let's see what we have here. Sine of negative X equals negative sine of X, pretty straightforward. These are things you just need to learn, all right? Along those lines, what else goes with sign? Cosecant, right? So the cosecant of negative X is negative cosecant X all right, cosine of negative X equals the cosine of X so in this case, it's the same thing. The negative really doesn't affect anything along those lines same with secant, the secant of X equals excuse me, the secret of negative X equals the secant of X all right, so when we have cosine and the negative inside, it comes out as positive when we have sign, negative inside, or it comes out as negative. Tangent of negative X is a lot like sign, it's going to come out as a negative. And think about it, sine over cosine. This has to work right because it comes out negative. It comes out positive, negative, a positive. It's going to make it negative, and that's what tangent is. So this also will work with co tangent of negative X is going to be negative cotangent X all right, so these are things you just need to learn, have it involved in your memory. All right, let's try some. Reduce these to a single trig expression. All right, well, this one's pretty easy. 

The sine of negative X, we already said, becomes negative sine of X, the cosine of negative X we already said comes the cosine of X positive. So a negative divided by a positive mean negative in sine over cosine that's going to be negative tangent of X all right? Over here we have a little bit trickier. Let's work with the negatives first. So tangent of negative alpha alpha is going to be negative tangent alpha and cosecant of negative alpha. That's like sign, remember, so that comes out. That's negative cosecant alpha. A negative times a negative is a positive, so let's now work with these. This is going to be sine alpha. Over cosine alpha times cosecant is one over sine alpha, these cancel. We're left with one over cosine alpha and reduce that down to what does that really? That's secant alpha. All right? So we reduce it, start with some fancy stuff, and reduce all the way down to a single trig identity. Now sometimes please understand sometimes when I ask you to reduce that down, you're actually going to reduce it down and get the number one, or the number two, all right? Just a single term. All right, so here is the identity, the Pythagorean identity, all right? This is a big one. 

Sine squared plus cosine squared equals one. Remember a squared plus B squared equals C squared. You can see that you do that on a unit circle. You're going to get sine squared plus cosine squared equals one. If you want to prove for it, Mister Bean would love to spend 20, 35 minutes on it, show you all the intricate details of it and just bore the pants off you. So please go ask Mister Bean for it, and he will do that for you. All right? And ask him, Mister Bean, can you show me on the board? All right? I need to see it on the board. All right, so here we go. Now, this alone is really powerful. In other words, anytime I see sine squared X plus cosine squared X, I know that I can plug in the number one. That's a big one. All right, one, we're simplifying identities and things like that. It's a nice thing to have. But there's another really great thing about this. I could subtract cosine squared and whenever I saw sine squared X, I could put one minus cosine squared X or likewise, if I saw one minus cosine squared X, I could put sine squared X same over here if I subtract sine squared X, I know that cosine squared X equals one minus sine squared X that's big, all right, or if I see one minus sine squared X, I can substitute cosine squared X so while this is the big deal right here, it leads to a lot of other possibilities, which are really phenomenal, all right? And it leads to a couple of other things. The first thing is watch this. We have two more identities. 

Now I'm going to give them to you. A lot of teachers don't give these to you, they think, oh, well, they can derive these formulas all the time. All right, they don't need to memorize them. Well, what's the point? All right? Let's derive them right now. I'm going to divide everything here by cosine squared X I can do that. All right? I'm going to drive everything, divide everything by cosine squared X, well sine squared divided by cosine squared, that is tangent squared X all right? Cosine squared X divided by two, that's one. And one over cosine squared X, that's the same thing as secant squared X so now we have another Pythagorean identity. All right? And you should probably write that one down. It's a big one. All right? So it is tangent squared X plus one equals sequence secant squared X all right? And along those same lines, if we had just tangent by itself, we could subtract one, right? We could have tangent squared X equals secant squared X minus one. So that's another thing we could substitute in. I see secant squared X minus one, I can put tangent squared X I'm giving you gold here, people. I'm gold. All right? 

There's one last one that comes from this Pythagorean identity. And again, a lot of those teachers out there, their academics, their higher thinkers, they're really smart guys. I totally appreciate that. Here's the thing. I want to make this manageable for you. Why would I make you do this every single time when the secret's out there? Here's the secret, right? Instead of dividing my cosine squared X, what happens is I divide by sine squared X what happens? Well, this is what happens. We get. Sine squared over sine squared is one. Cosine squared over sine squared is cotangent squared X and one over sine squared is cosecant squared X all right? So we have now really three big ones. All right? And then manipulations of those, we have plenty of others. When I say manipulations, again, cotangent squared X equals cosecant squared X minus one. All right, we have a lot of great information out there that we can use. So let's see how we can use this dough. All right, reduce this to a single trig ratio. All right, here's hot tip number one, write everything in sign and cosine. All right, well, I'm gonna do that. Tangent is sine theta over cosine theta. All right? Times cosine theta divided by cosecant is one over sine theta. All right, that's easy enough. 

So now let's see, oh, this is nice. These cancel, so now we have sine theta divided by one over sine theta, all right? When I divide by a fraction, that's the same thing as multiplying by the reciprocal. Sine theta times sine theta is sine squared theta. And there is our single trig ratio. All right? So I reduced everything to sign in cosine, did some fancy math for algebra. Actually, fractions 8th grade, 7th grade 6th grade, and I came up with my single trigger ratio. Now this brings up a fair point. X times X is X squared. Sine theta, or cosine theta, or whatever, tangent. When you multiply those together, we write that as sine squared theta. We do not, we do not write that as sine theta squared. It is not the same thing. It's sine squared theta. In your calculator, you may need to write that as sine, theta, group it all around, and square it. All right? But this is how we write it. Okay? That's really, really, really, really, really important note for you to understand. So myth number two, Pythagoras actually discovered the planet Venus, that is confirmed. It's true he was the first person recorded to discover the planet Venus. 

Pretty cool, actually. If you think about it, he didn't have a lot of the fancy technology that we have today to look at the stars. He was the first one to discover Venus. Way to go Pythagoras, who my boy. All right, verify the identity. Oh, we got another hot tip. Look for an identity. Oh, look Friday any. All right, well, here we go. Sine squared, my plus cosine squared equals one. Remember, oh, look at this. If I subtract this, sine squared is the same as one minus cosine squared. Okay, this is what we're talking about. This is an identity. It is equal to sine squared. I am plugging that in right now. Sine squared over cosine squared X, what's that equal? Sine of a cosine? Well, that is tangent squared X and that equals tangent squared X is which is we wanted. And there we have it. We have verified it. We have proved it. We are amazing. Let's take a look at the next one. Hot tip. Hot tip number three. Use algebra to make equivalent expressions. Oh my goodness. We love us in algebra. Here's what we're talking about. A lot of algebra we've been doing already, but this we can factor this. Sine squared, this is like X squared plus two X plus one. If we're going to factor that, that would be X plus one squared, or X plus one times X plus one. It's a perfect square trinomial, right? So let's write that with sine. 

So that is sine alpha plus one times sine alpha plus one. All right? And we're going to divide that by cosine squared alpha. Remember to keep in track here, we want to get it to one plus sine over alpha, excuse me, one plus 9 alpha, divided by one minus sine alpha. All right? So let's take a look here. What can we do? Well, again, we have cosine squared alpha. I remember that's my identity when I have cosine squared plus sine squared equals one. I know that if I subtract this, get cosine by itself, it's one minus sine squared, so I'm going to rewrite that. Sine alpha plus one times sine alpha plus one, and on the bottom, it's going to be one minus sine squared alpha. All right. Now this is difference of squares. It's like X's second -9. Difference of squares. That's X minus three times X plus three. Use an algebra again. So now here's something else. We've talked about this. Keep this in mind. Can I rewrite this side a little bit like that? I can. So I'm going to just rewrite this as one plus sine alpha, because I think that'll help us in the future. One plus sine alpha. All right. So this is difference of squares, as we talked about, the square root of one is one, the square root of sine squared is sine, so it's one minus sign, alpha, and one plus sine alpha. 

Oh, good thing we did right. We write that these cancel and it gives us what we wanted to prove. One plus sine alpha divided by one minus sine alpha and it does indeed equal the other side. That is a horrible alpha. I also apologize anyway, it is verified. We did our job. We used a lot of math in there. A lot of algebra, but we didn't no problem. All right, myth number three. Pythagoras was not actually the first person discovering to prove the Pythagorean theorem. This well-known theorem a squared plus B squared equals C squared. He was not actually the first person to discover that is actually plausible. There are many different instances where societies and cultures prior to Pythagoras being on this planet and they would write Pythagorean triples, three, four, 5, 5, 12, 13. They had that. They knew about that. So it is very plausible that he indeed, although he's named after him, did not come up with the Pythagorean theorem. All right, so let's take a next one, a tip, number four. Combine the fractions, combine the fractions. Let's see what we're talking about here. All right, I'm going to use hot tip number one. I'm going to rewrite all this in terms of sonic cosine. 

So this is cosine X over sine X and then we're going to multiply that by a cosine X, which is cosine X over one. Plus this is sine X over one, right? So this is just cosine squared X over sine X, we're multiplying plus sine X over one, and we want to keep in mind we want to get to cosecant X all right, now I don't always write this on the other side, and I apologize. That's just me being lazy. All right, so if I want to get a common denominator to add these, I need to multiply this by sine X so I have to multiply sine X on top and bottom. So now I have cosine squared X plus sine squared X that's on top. We have a common denominator of sine X that goes on the bottom. All right. Our identity alert identity alert, cosine squared X plus sine squared X, what's that equal? Oh yeah. That's a one. One over sine X and we know that one over sine X is indeed cosecant X there. We done did it. We verified it. It's all good. All right. All good all good. All right, myth number four, Pythagoras had a brotherhood almost a cult like following of students who went to him to study. And we're pretty sure this is what those guys look like. Pretty sweet, pretty awesome, right? Is that that is confirmed? All right? He had a whole thing. 

It was almost a secret society, all right? He would make people go for three years before they were actually really allowed in to his secret society, all right? Pretty cool guy, apparently he thought he was a little bit high on himself, so I guess, you know, all right, so bring the pain. Let's try this one here. See what we can do. We got a tangent squared minus sine squared equals tangent squared sine squared. All right, I'm gonna go over here. I'm gonna start with my very first tip. I'm gonna convert this to sine and cosine squared theta over cosine squared theta minus sine squared theta over one. All right. I am going to try and combine you. So I need a common denominator. I'm going to multiply this by sine squared or cosine squared theta, all right? So over here we now have sine squared theta minus sine squared theta times cosine squared theta, can't really do anything with that just yet. All over our denominator of cosine squared theta. And remember, we want to get to tangent squared theta and sine squared theta. All right now, something I notice here. I have this, I have a couple of terms and I have the same thing factor in both terms. To me, that means take it out. The greatest common factor. So sine squared, I'm going to factor out. 

So sine squared theta, take it out. So I have a one minus cosine squared theta all over cosine squared theta. All right? Now I'm going to separate these. I'm going to put this together over here. Sine squared theta over cosine squared theta. And I'm going to multiply that by one minus cosine squared theta. And why did I do that? Well, remember where we're headed. We want two things to multiply together. So I have one thing here, sine squared over cosine squared, which is tangent squared theta. Oh, that's great. And what do I recall? Sine squared plus cosine equals one. Sine squared equals one minus cosine. So one minus cosine squared right here. That's the same thing as sine squared. And boom, that is exactly what we wanted. Yay, us. All right? Verified. All right, this one is really it's had. It's had in the kitchen. If you can't handle the heat, get out, right? I want you to pause the video and try this one on your own, all right? All right, my recommendation to you right now is to listen to me, pause, write it down. Listen to me, pause, write it down. I've separated the steps so it's kind of easier to follow me that way. 

But this first step what I did was I made it all sign a cosine, then I got common denominators on top and bottom so that I could combine those fractions. All right? So this is just that step of combining them. Sine X times sine X is sine squared X, same thing over here, cosine squared X so on the top, I have sine squared X minus cosine squared X over sine X over cosine X, excuse me, alpha, and on the bottom, sky squared alpha plus cosine squared alpha. All right? 

Now, from the green to the blue step from this step, what I did was I just changed from dividing a fraction to multiplying by the reciprocal. And when I multiply by the reciprocal, I could cancel out the sign alpha cosine alpha here and here. I knew that sine squared alpha plus cosine squared alpha equaled one. So that left me with this right here. Sine squared alpha minus cosine squared alpha times one, which is just really sine squared alpha. Minus cosine squared alpha. 

Last two things I did was I took my sine squared and I change it using my Pythagorean identity to one minus cosine squared alpha minus cosine squared alpha, and there is just combined my terms. I have two negative cosine squared opposite, so one minus two cosine squared alpha. All right? I like trig identities a lot. There are a lot of fun to play with. There's no one right solution to do it. There's a lot of ways you could do it. You could make them a lot harder than I did. You can make them easier sometimes than I do. The key to remember is this. You got to show your work. You can't jump around the work is the answer, all right? The work is the answer you need to show that. So I wish all the best on these trig identities with the Pythagorean identities and I will see you on the flip side. Peace out