Here we go. We're going to be spending probably until around spring break about two plus months on trig. Most of you when I say trigonometry you think Sokotoa, sine, cosine, and tangent. That's how much trig that is. This is how much trig there is. This is how much we're gonna look at math well. What you think of as trig is really like saying drawing a circle is art. There's more to art than drawing one circle. Are you saying that trig is a form of art? Trig means three-angle measurement. Rotation angles can be measured in three ways. They can be measured in degrees or in radians. We're going to move to radians in a couple of days. We're gonna leave degrees behind. Radians, Trevor, are a much better way to measure angles for a number of mathematical reasons. There allows you a way to measure angles if you're trying to arithmetic in your head. How many degrees is this? Approximately 90. You know how many radians this is? Approximately 1.712. Not quite as convenient the number. Sorry. Such as life. But it has a whole bunch of other useful properties that we really really really really like. Okay. There is a third way to measure angles. They don't mention here. It's called gradians. GRAD. And if you have a scientific calculator, it probably had a DRG button or a degrees rad, rad button. Strangely, this is the one of the only things that your graphing calculators don't have and I haven't able to find it anywhere. A nerd that I am, I've looked. How many degrees in a circle? 360. Do you know how many gradians are in a circle? 400. They use it in engineering. Why is 400 nice? If there's 401 circle, how many gradians in this? 100, which means now you can describe an angle out of 100 gradians. Hey, that sounds like a percentage grade.  Now I had to look that up, Brett, because I'm only, I'm a nerd. I have a math degree. I did four years of university math. Never used gradians once. So, okay. I guess use them in engineering. That's the last I'm going to mention, gradians, but I had to mention it just for the completeness aspect. Angles can be measured in degrees where 360 degrees is one complete rotation and what we're going to try and do today, Eric, is we're going to try and link trig triangles, Sokotoa, to graphing because that's going to let us go in a general mode and that's going to let us expand what we could do with Sokotoa dramatically. A rotation angle is formed by having an initial arm that's fixed and rotating a terminal arm around a vertex. If you go in the positive direction, that's counterclockwise, so this way is positive. This way is negative. Positive angle comes from counterclockwise. A negative angle goes from clockwise. And Alex, if you have your initial arm on the x-axis, we say that it's in standard position. We're always going to start out in standard position. So example one says this. Draw the rotation angle in standard position and the angle they gave me is the rotation of 150 degrees. Trevor, if it's in standard position, I'm going to start right here. How many degrees have I gone? No, how many degrees have I just gone? Here. So far. 90. How far do I want to go? If I go all the way to this line, how far will I have gone? Too far. 150. About there. In fact, Emily, I actually know two angles. I know that this is 150. I also, with a bit of arithmetic, know that that angle right there is 30 degrees, which is really how I know to stop. B. Asks for a rotation angle. Do you have your book here? Did you find it? Yes. B asks for a rotation angle of negative 210. Mitsu, what's the angle that they want me to do? Negative means go this way. How far have I gone this way? Don't say 90. Fussy, fussy. Negative 90. How far now? Negative 180. If I go to here, I'm at negative 270. That's too far. In fact, how much further than negative 180 do I want to go to get to negative 210? Justin, how much further? In fact, I think I end up in exactly the same location. Eric, I end up in the same spot. You live? Okay. Eric, what's this last angle that they want me to draw? Positive or negative? Go this way. Eric, how far have I just gone? Look. Okay. Most of you know 360. Now, what I do, every time I go 90 degrees further, what I'm really doing is my nine times table. So you said 360, I think 36, 45, 450. How far? 54, 540? Too far. How far do you want me to go, Eric? About there. And you know what? As it turns out that the leftover angle is 30 degrees as well. In fact, all three of these have the same terminal arm. There's an infinite number of angles, Brett, that actually end up in this location. Because instead of going once around and stopping, I could have gone twice around and stop. Or three times around and stop. There's an infinite number of angles that will actually end up right here. Or I could go negative twice around, or negative three times around and stop, or negative four times around and stop. Angles that have the same terminal arm are called coterminal angles. These three angles are all coterminal. What's the correct answer? Well, they sort of all are. But there is sort of like a lowest terms answer. Don't write this down. Ellen, if I give you this, do I really want you to leave that as your final answer? How would I prefer you to write that as one hand? Same thing, okay? Well, what we have is something called the principal angle. It's the smallest positive rotation angle that's a long definition. Ryan, the principal angle is always between zero and 360. That's the lowest terms. In other words, these are all three the same angle is 150 between zero and 360. That's your lowest term. That's your principal angle. That's your principal angle. Okay. How can I find coterminal angles easy? Don't write this down. Just watch. Let's suppose that I went 140 degrees, making this angle right here 40 degrees. So my principal angle is 140 degrees because that's between zero and 360. How big is that angle? Can you find a mathematical way to get there without counting 90 by 90 by 90 by 90 by 90 by 90 by 90 by 90? Ellen, which is I think 500 if I do the math in my head correctly. Hey, how about add 360 twice? Or instead of adding 360, you know what I could have done? Minus 360 or minusing 360 twice or minus, you know what? Emily, I can add or subtract 360 as many times as I want. That's going to give us a mathematical expression for saying, here's all the angles. The mathematical expression is your principal angle and we're going to write this down in a second. Plus or minus multiples of 360. Don't write this down yet because that's way too much writing. How could I write multiples of 360? Well, instead of writing multiples of 360, I'm going to say plus or minus 360 times a number, times one, or times two, or times three, or times four, or times five, or times six, or times seven, or times eight.  Am I allowed to put in decimals?  But I can also put a negative number here. So instead of going plus or minus, I could just go put a negative number there. I'll have to go plus or minus. Here's what we write. We say where m is an integer. Now, you guys have seen this all reals before. Now, we're saying this time it's not all reals. This time it's all integers. What are integers? Not positive or negative numbers, actually. Positive or negative non-decimal whole numbers. One, two, three, negative one, negative two, negative three are integers. Negative 3.6 is not an integer. So the principle, if I want to find every single angle theta, it's the principle angle plus multiples of 360 degrees where n is an integer. It's easier to do than to write.  It's really easier just to do it. Next page. Example one. Find the angle between negative 360 and positive 360. That's coterminal with 200. Okay. You know which angle they want? That one. How big? It's going to be the principle angle that they gave me. Am I going to add 360 or minus 360 to get that one? Minus 360, which is what? Which is what? I think, isn't it negative one four? No, no, it is negative 160. I stand corrected. Mr. Dewey, bad math. This angle right here is negative 160 degrees. It's coterminal to positive 200 degrees. Yeah, never the same spot. What about, oh, coterminal with negative 290 degrees. So that's negative 290. I want this one. How will I find the coterminal angle to negative 290 degrees? Specifically, the angle is between zero and 360. Yeah, so between negative 360 and positive 360. I didn't hear. I heard negative, which one? Either of those would give you a coterminal, but since they want my answer to be between negative and positive 360, I think I have to add 360 to this one, which gives me what? 70 degrees. Which of these two here is the principle angle? Which of these two here is the principle angle? Which of these two here is the principle angle? It's meant to be really easy. 200. Which of these two here is the principle angle? 70. Why? Because the principle angle is between zero and 360. It's the smallest positive one. It's the lowest terms equivalent. Then they ask us to write an expression which represents all of the angles. Okay. For the first one, all of the angles are going to be the principle angle plus or minus multiples of 360 degrees, where n is an integer. Now, Tyler, I was taught actually to write plus or minus, and I still will sometime, but a few years ago, they said, hey, look, if n is an integer, you're introducing a negative. You can also introduce a negative right here, which means the minus there is kind of redundant. And I said, okay, fair enough. So in the book of the book, they probably won't put the negative right there. I often will in my notes or in my hand. If I'm in a rush, you'll see it. I don't take marks on this. Either is valid. How could I write an expression for all of the angles for this second one? It's the principle angle plus multiples of 360, but the multiples of 360 have to be whole number, positive, or negative multiples, multiple. Example three, it says, for each of the following angles, determine the quadrant of the terminal arm and the principle angle.  This is quadrant one. This is quadrant two. This is quadrant three. This is quadrant four. Traditionally, we use Roman numerals to label the quadrants, and you do need to know, you need to memorize which quadrant is one. It starts with positive, and it goes in the positive angle direction all the way around. So having said that, for about 90, for the rest of this unit, oh, for about two months now, about 80% of the time, the first thing I'm going to do, my physics 12s, I know this is dull. What does dull stand for, physics 12s?  I guess I could say dog. They want the quadrant. Here's 265. I'm going to go like this. I'm going to start a SAR in standard position right there, and then ready a SAR. We're going to go for a walk. Here we go. Here we go. Here we go. How far? How far? If I went all the way to here, how far? 270. How far do they want me to go? Just shy of 270. In fact, I can tell you exactly how shy, five degrees shy, but they didn't ask me for that. They asked me, what quadrant are we in? What quadrant are we in? Quadrant three. What's the principal angle? Now that's the angle, the smallest positive angle. That's the angle between zero and 360. In this case, it's kind of a trick question. You know what the principal angle is? 265. It is the principal angle. Eric, you know what I'm going to do first? They're right. Maria, what angle did they give me? No, they didn't give me 111. That's hugely important. What angle did they give me? You said negative? That means I'm going this way. How far have I just gone? Now I haven't gone 90. Can't be doing that. How far have I just gone? Negative 90. If I go all the way to here, how far will I have gone? Too far, which means I'm in there somewhere. Care of what quadrant? Is negative 111 my principal angle? No. In fact, you know what is? That guy there is my principal angle as a picture. How can I calculate it without drawing a picture if I'm lazy or if I just want to calculate it? It's going to be negative 111. What? Plus 360, which is what? 249? Ready, Alex? What's the angle that they gave me? How far? How far? Now here's the trick. Use your nine times table. So 36, what comes after 36 and your nine times table? 450. What comes after 45 and your nine times table? 540. What comes after 54 and your nine times table? 630? Yes. 720. Now some of you might have been able to go twice around as a 720. Some of the basketball players might not have heard that term rate. That's fine. What comes after 720, Alex? 810? How about here? Yeah, 981 plus 9900. How about here? What comes after 900? 990 or 99, right? So nine times table. How about here? 1080. And what do they want me to draw? I meant 1080. That much further. There's my angle. Alex, what quadrant? Yep. What's the principal angle? Now there's two ways to find it. I'm pretty sure the principal angle is that I think it's five. But if I didn't have the graph in front of me, Ryan, I could take the angle that they gave me. I could minus 360. Am I done? Am I between zero and 360? Is this the smallest possible? Minus 360 again. Am I done? Is this the smallest positive angle between zero and 360? Minus 360 again, and now I reach math that I can do in my head, Spencer. What's 365? Take away 360 now that you're awake. Yeah. There's the principal angle. Another thing you learned with triangles was Pythagoras. Pythagoras is a squared plus b squared equals c squared. But what we want to do today, Tyler, is we're going to superimpose this triangle onto a graph. I'm going to tell you that this point right here is x comma y. It's on graph paper. And if it's x comma y, what that also means, Asar, is this distance right here, this distance right here, or this length right here is x. In other words, if you're 3 comma 4, you're 3 over. And 4 up, this distance right here is y. In other words, if they give me a point, what they're also giving me always from now on, Mitsu, is part of a triangle. And because we're rotating around in a circle, instead of calling this a hypotenuse, we're going to call this r. I went to go see a pirate movie the other day, but I couldn't get in because it was rated r. What does that mean? It means Pythagoras. This means, Eric, that Pythagoras can be written as x squared plus y squared equals r squared. This is hugely important. Highlight it. You need to memorize this. You need to memorize this. You need to memorize this. You need to memorize this. You need to memorize this. In fact, Trevor, this is so important that in a few minutes, I'm going to say, repeat after me and you're all going to say x squared plus y equals r squared. We're going to do that right now. We're going to chant it together. Are you ready? Repeat after me. x squared plus y squared equals r squared. Eric had no idea what was going on, so we're all going to have to do it again. Are you ready? Repeat after me. x squared. Justin never moved his lips at all, so he's going to solo. Justin, repeat after me. That's very profound. You don't know it yet, but it is. Not only is x squared plus y squared equal to r squared, get the x by itself.  Justin, how would I get the x by itself? No, I would not square root. I would minus the y squared. I would get r squared minus y squared equals x squared. Then what would I do? Square root. The square root does not cancel out those squares. Get the y by itself, Justin. I don't have these two memorized. You can if you want to, Tyler. I just rederived them. I know this and I can get any letter by itself. By the way, this gives me r squared. How do I get the r by itself technically? Square root. x squared plus y squared equals r squared and x equals the square root of r squared minus y squared. y equals the square root of r squared minus x squared. Terribly profound. Next page. I picked the physics 12 student because we've been doing trig in physics 12 like crazy. We have what are called the primary trig ratios. Now there's something called the reciprocal trig ratios. You're going to learn three new trig ratios today. But the primary ones are the ones that you did in grade 10. In terms of opposite adjacent and hypotenuse, sine is what over what? My physics 12s are like, oh, for Pete's sakes in my sleep, opposite over hypotenuse. Yes. And cosine is what over what? Adjacent over hypotenuse, which we write as a over h and tangent is what over what? Opposite over adjacent. And for 100 years or so, probably more than that, English speaking students have used this acronym right here, SOCA TOA, to help them remember that sine s is opposite over hypotenuse. See the opposite over hypotenuse in there, Ryan? And cosine is adjacent over hypotenuse and tangent is opposite over j. Eric, there are three new trig ratios. By the way, it's sine, not sin. Sin is when you swear in my class or are late. That's a sin. Sine is the trig function. Cosine and tangent. Although often we abbreviate tangent as tan and cosine as cos. We don't abbreviate sine as sin. Sine is sine. There's something called the reciprocal trig ratios. The reciprocal trig ratios are cosecant, secant, and cotangent. We'll pause for a second. So let's keep going. Trevor, the reciprocal trig ratios are cosecant, secant, and cotangent. And they're actually easy to remember. It sounds confusing. It's not. Cosecant is the reciprocal of sine. It's one over sine. So if sine is opposite over hypotenuse, what's cosecant? Hypotenuse over opposite. If secant, abbreviated sec, is one over cos, what is secant in terms of opposite adjacent and hypotenuse? If secant is one over cos, what's secant in terms of opposite adjacent and hypotenuse? Kara. Yeah. Cotangent is one over tangent. Okay. What's cotangent in terms of opposite adjacent and hypotenuse? Adjacent over opposite. I don't memorize these. I memorize, okay, really and truly, put the report cards away. There's nothing on there significant anyways. Thank you. So how do you know which goes with which? First of all, you need to memorize that cotangent goes with tangent. Is that too difficult to memorize? I'm going to suggest that it's fairly easy to remember that cotangent goes with tangent. For the other two, what everyone remembers is cos never go together. Cosine does not go with cosecant. And sine goes with cosecant. Cos never go together. So I don't memorize what secant, cosecant and cotangent are in terms of hypotenuse. I just know if you say to me, hey, what's cosecant? I go cosecant is sine, sine is opposite over hypotenuse. Hypotenuse is over opposite. It takes one second and I can do that in my head every time. And that's actually the truth. Usually when I say to you guys, I haven't bothered memorizing this, I'm lying because in 15 years I have memorized it after teaching all this. This I haven't bothered. Not worth my time. Not even worth making an effort. So you can remember it. Each pair, come on, come on. Each pair has only one co in it. And now, for the most profound moment in the next two months, we can also rewrite these in terms of x and y and r. What we can do is take this triangle right here and put this part of the triangle at zero, zero on a graph so that this is still opposite and this is still hypotenuse and so hypotenuse, Mr. Dewick. Good gosh. Adjacent and this still is hypotenuse, but instead of saying opposite, Asar, I can say what? Because I asked you to what? Oh yeah, that joke is back with a vengeance this unit. Why? And instead of saying a, what can I say? Adjacent is actually, and instead of saying hypotenuse, what can I say? These are your new definitions for trick. Sign is still opposite over hypotenuse, but in terms of a graph, sign is what over what? In terms of x and y and r, sign is what over what? Cosign is what over what? Tangent is what over what? You're not getting it. Ellen, see this right here, this sentence beginning with the word the? In just a second, I'm going to ask you to read it in a nice loud voice. Are you ready? Read it out. Under enlightening should go right now. It's profound. There should be a deep rumble in the distance. You have no idea just what this has opened up for us. The fact that from now on, sign is no longer or not only opposite over hypotenuse, it's y over r. Cosign is x over r. And tangent, there should be a breeze blowing through this classroom right now. Your hair should be ruffled and often the distance you should hear violins or something like that. It's profound. Ellen, could you read this to me? I'd like you all to cross out the word should and instead write, what does all caps mean in an email? So much so that for the next oh, three or four weeks, I'm going to start every class by saying, Mitsu, sign is what over what in terms of the graph. Alex, cosine is what over what in terms of the graph. Ellen, tangent is what over what in terms of the graph. Too slow. It has to be at your fingertips. Now you'll notice, Trevor, have I filled these ones in? I don't have them memorized. You know what I do have memorized? I know that cosecant is the reciprocal of sign. So if sign is y over r, Trevor, what's cosecant?  And if secant is the reciprocal of cosine, Justin, what's secant? And if cotangent is the reciprocal of tangent, Maria, what's cotangent? I re-derive those every time in my head. In fact, whenever we're doing a secant or a cosecant or a cotangent question, you'll see me pause for a second. What I've just done is I've associated it with its primary trig ratio. Ask what's the primary and flip it. Now, there is one easy way to remember one of these. Eric, can you read to me this one here? No, the whole thing. Physics 12s are going to hear something familiar. Sine is y over r, but it's not y over x. You have to memorize these. In fact, I'll say if you don't have these memorized, you'll flunk the test and you'll be dropping out a math flood. Now, having said that, Mitsu, almost all of you, it'll happen naturally because you're going to be writing this probably five or six hundred times in the next couple of months. But what I am saying is, if in about three weeks you're hesitant on this, you in trouble. You better sit down and write it out 50 times each. Then you'll have it. Why is this so profound? Why should thunder and lightning flash next page? Example five says the point 15 comma 8 lies on the terminal arm of theta as shown. Calculate the value of r and hence determine the exact values of the primary and reciprocal trig ratios. What this question is telling me is that we can find sine theta, cos theta, tan theta. Which reciprocal trig function went with tangent? Let's write it right opposite like that, cotangent theta because that helps me stay organized. Which one went with cosine, not the code? Which one was not the code? Secant, sec. Which one went with sine, the code? Cosicant. We're going to fill in all six of those. What have they given me here? An x and a y. What letter is right there? What's the first thing they want me to calculate? R. Did I have you guys a few minutes ago chant an equation that had x and y and r in it, oh pray tell? How are x and y and r related? Someone say it out loud, sure. Someone besides Kara, well done by the way, I'm impressed.  In this question, what's x15 squared plus, in this question, what's y? 8 squared equals r, oh Justin, how do I get rid of the r squared? Okay, I'm going to put a big square root over here. This does not cancel out those two squares. You get r equals the square root of, what is 15 squared plus 8 squared? 15 squared is 225, 8 squared is 64, 225 and 64, 289? What is the square root of 289? Does it work out evenly? Oh, so you know what r is? 17. Sine is what over what in terms of x and y and r. If you have to turn back and look, go ahead, but sine is what over, I've heard two different answers, I need the right one. y over r, specifically here then, what's it going to be? What over what? And off in the distance, there was a rumble of thunder. Oh, and now I can do cosecant right away as well, 17 over 8, yes? That's why I made sure to write them in their pairs, it makes the second part way easier. Cosine is what over what in terms of x and y and r. So specifically for this one, what's it going to be? And secant will be 17 over 15. Tangent is what over what in terms of x and y and r. y over x, specifically for this question, what will it be then? Do you realize what we just did? We got all six trig ratios and we don't even know the angle. And we didn't need our calculators to do it. Didn't hit a sine button. Didn't hit a cosine button. Didn't tangent button. We calculated it. Fairly easily, in fact, I would argue if I had given you smaller numbers here, this is completely fair game as a non-calc question. As a matter of fact, folks, this next test is going to be all non-calc. Think about what we've just done though, Mitzi. This is why linking graphing and trig, this is why thunder and lightning should roll. We were able to do trig, sine, cosine, tan, no calculator, not even finding the angle. I like example six. I like example six. I like example six. I like example six. So example six says this. Angle A terminates in the first quadrant. You know what? I should draw that. There's angle A with the sine of A equaling one-third. Sketch a diagram, already did, and then find the other five trig ratios for angle A. So we know the sine of A equals one-third. We want to find cosine of A, tangent of A, which reciprocal goes with sine, cosecant? I guess this one I can already do, three over one. What goes with cosine, secant? And what goes with tangent, cotangent? What did they tell me the sine was here, SR? Sorry, you can't say it louder. Sine is also what over what in terms of the graph, in terms of x and y and r. I think what they've really done is they've told us one over three equals y over r. So based on that, what's y have to be? What's r have to be? Three? What's x have to be? Oh, I don't know. It would be wonderful if there was some kind of an equation that related x and y and r, that maybe we get chin. Is there? What? Spencer get the x by itself and then r squared minus y squared to the other side. Square root. What was r squared? I think that's fair game as a non-question. Put your calculator away, Spencer. Minus. What was y squared? I think asking you to go nine minus one in your head is completely fair game. What is nine? Take away one. We want you to leave it like that. I don't want decimals. I want exact value as well. I don't quite want you to leave it like that. We're actually going to ask you to write it as two root two.  You're going to be asked, Joel, to simplify square roots. For example, if I tell you the answer is root 20, 20 is what times what? Now two times 10 is not what I'm thinking. 20 is what perfect squares times something. 20 is what times what? 20 is root four, root five. What is the square root of four? It's math 10. Hey, what about root 40? What times what? I know it's five times eight.  It's what times what? So root 40 is root four, root 10. What's the square root of four? Two, root 10. What if they give you something like this? Root 75. What times what? Now I'm going to, Ryan said three times 25. I'm going to say it the other way around always. Perfect square first because I'd like to write it this way, Ryan. Root 25, root three. Doesn't that look nicer because what is the square root of 25? See, I get my five root three popping out of here. You guys remember doing this in math 10? Now I expect my math 10 kids to be able to do this in their heads. I push my kids a little bit.  There comes a point when I say, for Pete's sakes, find the perfect square that goes into it. Take the square root and you're done. Root 80 is what times what? Eight times 10 doesn't mean no good. Those aren't perfect squares. Now a lot of kids will say root four, root 20. Which is fine, Emily. What's the square root of four? Here's my question. Is 20 as low as it can get? Because 20 is what times what? This is going to be two, root four, root five. What's the square root of four? This is going to be a two and a root five. Hey, what do you think I do with that two and that two? Now some of you who are better at your time tables than Emily, sorry, not 16, Mr. Dewick. Some of you who are better at your time tables than Emily might have noticed 80 is 16 times five, which gets you there in one step, but you don't need to get there in one step. You're going to hop Scotch your way there just fine. That's a really quick review of simplifying square roots, but that's where the two root two came from. Let's root four, root two, take the square root of two. Okay, here we go. Cosine is what over what in terms of x and y at r. x over r, more specific for here since x equals two root two, two root two over three. And this is going to be three over two root two. Think about what we're doing here. We're finding exact values of trig functions and we still don't even know the angle. We don't care. By linking, graphing and trig, all we need to know is where the terminal arm goes through. This is how they generated a bunch of the old trig tables, by the way, in case you ever wanted. What they do like 200 years ago before calculators, how do they get trig tables? Tangent bret is one over what in terms of x and y and r. Good recovery. I wasn't sure because you were out of the room. I was going to see you. Oh, so here specifically, what's it going to be? Three is r, one over root two, one over two root two and two root two over one. In fact, you know what? I feel embarrassed that I wrote the over one up here too. It's like washing my hands afterwards. Thunder, lightning, bumbling. Also, we're going to do some standard what you guys have all thought of as trig. Using Soka Toa to find ratios, angles and sides. Somewhere along the way, you're probably going to get a triangle with a missing side. Here's my angle. Lad, never mind, you're not physics 12, you're physics 11. Emily, opposite of Jason, you're hypotenuse. How about the m? Thank you, m for the m. Mr. D, which trig function? Yep. What always goes next to the trig function, the angle, which in this case is 37, equals whatever what Emily, and here just once I will put that over one, even though it makes me feel like I've washed my hands, because I can solve this how much cross multiply. In fact, I think you're going to get m equals 83 divided by cosine of 37. Try this, all of you on your calculators, because this is also the time for you to find out if your graphing calculators are in the light mode. Your graphing calculators by default, if you've done a reset or a battery swap, they tend to go back to radians. So you want to just see if you get 103.9, if you don't, you're in radians. Anybody not get 103.9? So press your mode button right here, and then make sure degrees is highlighted, not radians. To highlight it, just press enter on top of it. And you will notice, as I said, it has radians, it has degrees, it does not have radians. I'm quite surprised by that actually. Not only can you use trig to find sides, you can use trig to find angles. So it says, in example eight, use a diagram to determine sine A and the measure of angle A to the nearest degree. Okay, sine is what over what in terms of opposite adjacent hypotenuse? Opposite over hypotenuse, if that's angle A. I think it's opposite 29 over hypotenuse. Sine A is that. How do I find the angle? Now my physics 12 shush, you've been doing this all year long. How do I find an angle? If I know that the sine of A is 29 over 34, well I do. Y'all remember? K physics 12, what do you do? Inverse, second function, shift.  It appears there. It's saying the inverse sine of 29 over 34, the angle is 58.5 degrees. Oh, it says to the nearest degree, angle A equals 56 degrees. 56? Tara was giving me a bit of a frowny face. How about 59 degrees? Yes, thunder, lightning. What's your homework? Folks, all of it. 1 to 11. You need to practice this. Now, I am not going to give you new homework on Friday, so ideally you'll be able to get this all done before your Christmas break. I'll touch on a few things on Friday, but we'll also have some math nerdy fun, I think. So homework, 1 through 11.


The covert narcissist will control your mind. They will seek the power to control how you think and perceive things.  They believe that by controlling your mind, they can then control your behaviors. They can then keep you around them as their source of supply. Covert narcissist have a really strong need to be in control. They need to influence and direct your behavior at the course of events. They lack empathy. They have no consideration for how their actions affect you because they're trying to exploit you. They're trying to make full use of and derive benefit from you and they will try to extract what they want from you any way that they can. They can be very manipulative. They can be very scheming and calculated. They have this alternate reality where everything has to be interpreted their way. Everything has to be done their way, which is all based on their distorted thoughts and perceptions. It's all based on trying to protect themselves because covert narcissists are very insecure, which is why they have such a strong need to be in control. They have to be superior. They think that by being the one in the dominant position, they can then protect themselves from danger or harm. They can then ensure that everything pans out in their favor. The covert narcissist doesn't always need to be the center of attention. They can be more relaxed and cheerful. They can blend in well. They can seem friendly. They can even seem nervous or shy. Until you notice something that is gradually developing where they are trying to create confusion within you because they want you to feel confused. They want you to doubt what you believe to be true. They want you to doubt your own memory, perception and sanity. It's their way of gaining superiority over you. It keeps you locked under their control. Covert narcissists are self-absorbed. They are very exploitative, but you may not realize this at first. When you first meet them, they can seem like they have a personality, but you soon discover that they lack depth. They can be very shallow and superficial. They have a lot of black and white thinking. They think they're superior to everyone, but they won't openly reveal this to people because they know that not everyone is going to agree with it. Not everyone is going to see them as being superior, and they are very sensitive to criticism and insults. So they don't want to put themselves in a position where they are at risk of being criticized. They are very unreliable. They are very untrustworthy. You cannot depend on them for anything. They will always let you down. Even if you agree with them on something, they will still go and do it their way because they just don't care about what is right for you. Everything has to revolve around them because they always think they're right. They always think their opinions and ideas are better. Although they may not reveal it to you, when you are around a covert narcissist, you will always feel like you've done something wrong. You will always feel like you've done something to hurt and offend them because they will always be upset with you and you will always feel like you need to do something to fix the situation because they're gaslighting you. They're sowing seeds of doubt within you to make you question your own memory, perception and judgement. They're trying to keep you guessing about yourself. Covert narcissists love to play the victim role. If you have your own thoughts, ideas or preferences and you disagree with a covert narcissist or you change your mind about something rather than trying to understand you, the covert narcissist will see it as though you're trying to hurt them as though you're trying to make their life difficult because it's all about them and if you're trying to live your life your own way in their mind you are doing something to them they will play the victim role to make you think there's something wrong with you.  They want you to think that you're always a problem in every situation. Whenever you do something good for them they will always minimise their gratitude and appreciation. If you do go out of your way to do a favour for them they might say thanks but there's never any emotion to it. You never feel like they really appreciate anything you've done for them. They always have to minimise everything you do. They have to minimise their gratitude and appreciation. You never get the feeling that they really value anything you've done for them because they have a lack of empathy. They can't really share the experience with you or they don't want to give you the satisfaction they would rather stay disconnected from you because it always leaves you questioning yourself it always leaves you wondering if you did something wrong and that is how they gain control of your mind. They will complain that you never do anything together even though you may be constantly offering to do something with them and then they will turn you down on your offer only to then come back and complain that you never do anything with them. They will say that you spend too much time with yourself they will say that you're selfish even if you've offered to spend time with them because they want you to question yourself they want you to feel like you're doing something wrong they minimise your achievements and positive experiences whenever you try to share something with them they always act like it's not a big deal they will say that someone else has done something better or they will quickly change the subject because they don't want you to realise your worth they don't want you to realise how great you are no matter how great a situation may be they can always find something wrong with it it keeps you looking at yourself and questioning if maybe it wasn't that great and you always feel like you've offended them because that's exactly how they want you to feel the COVID narcissist is always trying to gain sympathy there isn't a day that goes by where there isn't something happening to them there's always something wrong they've always got something to complain about and they will always expect you to listen to them as they rant and rave about everything that's not right but when you listen to what they're saying you realise they're just normal things that everyone deals with and either if you try to offer your support or encouragement they will always leave you feeling like you don't understand because they don't want to resolve the situation they like having problems they like having something to complain about it gets them attention and it helps them to gain sympathy the COVID narcissist will not enter into your experiences they will avoid any of your situations that would naturally evoke an emotional reaction maybe something went wrong for you maybe someone died and the narcissist may acknowledge the situation but they won't follow it up with any questions they won't show any curiosity into how you feel about it they won't care about how it's affecting you or even if it's a positive experience they won't enter into it with you they will just relate it to something that happened to them they always minimise your experiences because they have a lack of empathy it always leaves you feeling like they don't care about you as though you're not significant which then keeps them feeling superior to you it keeps you under their control when you get involved with a COVID narcissist they will only ever try to control your mind they don't want to have a good relationship with you because they fear being vulnerable no narcissist is going to be able to be vulnerable with you they cannot connect to you they cannot talk about their issues they are fear based and shame based people but if they can gain control of your mind and keep you confused they can then project those emotions onto you and watch you going through it instead of dealing with it themselves there's no point questioning or confronting the narcissist they're just going to see it as another opportunity to gas like you they want you to doubt yourself but instead of trying to get them to validate who you are you need to recognize that you have a good character you have good intentions and then separate yourself from the narcissist when you are involved with a COVID narcissist you don't realize it until it's too late but once you realize what you're involved in you need to distance yourself you cannot save them they're always going to be this way but you can spend time with people who are emotionally healthy your life doesn't have to revolve around confusion thank you for watching I hope this video isn't with you please like, comment, share and subscribe if you would like to donate my paypal link is in the video description coaching inquiries you can email me at coaching.nrcsurvivor.uk thank you for watching and I'll talk to you soon


Just a similar package, $1.9 trillion has passed the House of Representatives. $1.9 trillion, which includes a $1,400 check to most of you, and maybe I'll get one, who knows. It also includes a huge amount of money. There's health for small businesses through the Paywall Protection Program. There's a massive amount of money that goes to state and local governments. There's a massive amount of money that just goes to a variety of different projects and different programs that have nothing to do with COVID. I mean, this is all built as a stimulus plan, but there's nothing stimulating about it other than handing people cash. Most of it is just throwing money away. Most of it is just growing government spending, growing government power and printing a lot of money. A lot of pork, a lot of pork, and there's going to be more as Biden works his way through his agendas. This is going to be a pork filled legislative Congress. Now, beyond that, the House stimulus package included an increase of $15 an hour to the minimum wage. All that would slowly increase and kick in in 2025. I've talked a lot about the minimum wage, and I'm going to talk about it again. I mean, it's an awful idea, really, really bad, really, really bad in particular for those that it pretends to help, that it lies to us about helping. But the thing about the bill is that the Senate will not vote for raising a minimum wage. The Senate has ruled already that it cannot vote as part of the package, so it would have to have a separate vote on the minimum wage to get that passed. They would have to have 60 votes. They don't have 60 votes. It's likely that they don't even have 50 votes, because at least two Democrats have said that they do not support raising the minimum wage to $15 an hour. The progressive part, the progressive part of the Democratic Party are flipping out. They're going nuts, because they think that this is an integral part of the agenda of the left, and it has to pass, and yet many Democrats oppose it. So it's just not going to happen the $15 minimum wage. Of course, Republicans, as always, are coming to the rescue. And so Mitch Romney is proposing a $10 minimum wage, and Josh Hawley is proposing instead of a minimum wage some income supplementation. So there's now a call among Democrats to unite with Romney and potentially with Josh Hawley to pass some kind of bill to raise wages of employees. Anyway, a total disaster. Instead of the Republicans holding the line, opposing raising the minimum wage unequivocally, and they would win that battle, they're already starting to negotiate and starting to compromise. So who knows? They might have a filibuster proof 60 votes to raise the minimum wage, just maybe not to 15, maybe to a lower number, but there's a good chance it will be raised. So thank you, Republicans, for being such a great opposition party. What we need today, what I called a new intellectual would be any man or woman who is willing to think, meaning any man or woman who knows that man's life must be guided by reason, by the intellect, not by feelings, wishes, wins or mystic revelations. Any man or woman who values his life and who does not want to give in to today's cult of despair, cynicism and impotence, and does not intend to give up the world to the dark ages and to the role of the collectivist. All right, before we go on, reminder, please like the show. We've got 163 live listeners right now. 30 likes. That should be at least 100. I think at least 100 of you actually like the show. Maybe they're like 60 of the Matthews out there who hate it, but at least the people who are liking it, I want to see a thumbs up. There you go. Start liking it. I want to see that go to 100. All it takes is a click of a thing, whether you're looking at this, and the likes matter. It's not an issue of my ego. It's an issue of the algorithm. The more you like something, the more the algorithm likes it. So if you don't like the show, give it a thumbs down. Let's see your actual views being reflected in the likes. But if you like it, don't just sit there. Help get the show promoted. Of course, you should also share and you can support the show at your own book show.com slash support on Patreon or subscribe star or locals and show you support for all for the work, for the value. Hopefully you're receiving from this.  You'll know what shows are on when they're on. You'll get notified. So yes, like, share, subscribe, support, like, share, subscribe, support. There you go. Easy. Do one or all of those, please.


Murphy could I have your dog now and you're welcome to him Chester if you can find him what and he heard some wolves and run off for me on the way out here I said let's get out of here Chester the awful knot on your head did I get you into all he done everybody else first I think I better go clean up the cells section pal kindly tore up a chair and a mattress out there won't take me long rest of chicken tell by his hat boy comes from how he makes his living well next week a man dies because of a difference of opinion transcribed story on the United States armed forces radio and television service


Oh Dashing through the snow in a one-horse open sleigh over the hills we go screaming in terror all the way because crampus is chasing us Welcome to another episode of spooky santa. That's me. I'm santa Some people call me father christmas, but in this episode I'll tell you a spooky story about father frost He's known in Russia as Barosco Father frost he rewards those who are kind and polite and he punishes those who are not That sounds a lot like what I do doesn't it? Oh, but father frost. He's not nearly as nice about it as I am I also received an email from a mother in New Mexico and she tells us what her three-year-old Son told her once that creeped her out And if you want to write a scary story of your own you can email it to me at letters at spooky santa comm But first I have a story about a young girl who receives an evil doll for Christmas It's called purple fingernails now bolt your doors lock your windows turn off your lights Pour a mug of hot cocoa and come with spooky santa for another holiday chiller There was once a little girl who had a large collection of dolls One Christmas she saw a doll in the window of a department store that she Absolutely had to have it had long flowing blonde hair and a lovely blue dress But the thing that impressed her the most about the doll was that it had lovely bold purple fingernails When Christmas morning came she was delighted to find the doll with the purple fingernails lying under the Christmas tree She played with it all day and when she went to sleep. She put it on the table at her bedside That night the little girl was awakened by strange noises. It was still dark outside She looked over for her doll and it was nowhere to be found The girl puzzled got out of bed and walked into the hallway The door to her parents room was a jar and she pushed it open softly and looked in Her parents appeared to be sleeping soundly. So she went back to bed In the morning she saw the doll with the purple fingernails It was sitting on the table by her bed exactly where she had left at the night before She assumed that she'd just been having a bad dream But when she went into her parents room that morning, she found them both Lying dead in a pool of blood. There were two purple doll fingernails stabbed into their hearts It's a short story, but it definitely is a scary one But not nearly as scary as something that really happened to Sarah We get email we get your email every day Here's your mail today I love receiving your emails. You can email me anytime at letters at spooky santa.com I particularly like receiving scary stories that you write Either original stories that you make up on your own Or stories that really are true something that happened to you Like what Sarah sent me Sarah lives in albuquerque new mexico and her son is nine years old But she sent me an email to tell me about something that her son said when he was only three years old And it is kind of creepy.  It was hard to understand him One day while I was driving he starts telling me a story and it started Mommy when I was in your tummy. You worked at McDonald's. I said, yeah Wondering where he had heard that since my stint there was very short And then you got in an accident and were hurt Yes, I was buddy, but it all turned out. Okay. You were fine and safe in my tummy Well at this point Sarah was weirded out because even though she was still dealing with lawyers regarding the car accident She didn't think she had ever told her three year old son about it At least not in detail and never about McDonald's. He continued and said, I know mommy. I was watching over you What she said quickly Yeah, I was watching over you next to a man in white. He said you were going to be okay After that I didn't get another response out of him. She says he's now nine and remembers none of this Oh, that is creepy. But also kind of inspirational It might mean that angels really are watching over us and maybe we get to watch over those we love even before we're born Very good story. Sarah. Thank you so much for sending that to me Now for our third story and the last one of today It's called Frosted by Ali mathews and it talks about father frost who in russia is known as morosco Here's the story Once the stockings were hung on christmas eve instead of reading the night before christmas or Root off the red-nosed reindeer like normal people They had to listen to the story of a creepy russian jack frost wannabe who froze people Logan groaned as his grandmother who made them call her babushka Because that was the russian word for grandmother launched into her yearly retelling of father frost Though he didn't miss the look she shot him. He refused to be intimidated by his grandmother He was too old for her silly fairy tales He knew the story well enough to tune out while it's still appearing as if he was listening The torture was worth it as long as his new gaming computer was waiting under the tree for him in the morning At 10 years old his younger brother tanner was easily drawn into the story Here's what grandmother said A few years after his wife died an old man married a woman With a daughter of her own and it soon became apparent that his new wife had no use for his daughter I no longer wish to look upon that horrible girl. Take her deep into the snowy cold forest and leave her there The old man did as he was told for his wife always got her way He dumped the poor girl in the woods where she sat beneath a tree Shivering with cold her hands and feet growing numb Soon father frost sailed across the treetops the branches and twigs crackling and snapping as he moved above her After settling in the branches above her. He asked are you warm enough my lass? Despite her chattering teeth. She said I am very warm. Thank you father frost Tanner's eyes widened. He was so gullible Grandmother continued her story Father frost moved closer still the branches snapping and popping in his wake Are you warm my innocent lass? He asked again The girl's lips were stiff with cold, but she forced them to move. Yes dear father frost. I am warm enough More pops and snaps erupted as father frost descended to the ground He leaned over the girl. Are you warm my sweet lass? The girl's fingers would no longer move, but she squeaked out a response Of course dear father for father frost. I am warm enough Mother frost took pity on the girl and wrapped her in warm furs Watching the story Tanner let out a sigh of relief and Logan rolled his eyes They'd heard this story a million times.  He was almost 14 years old Grandmother continued with the story In the morning the old woman ordered her husband to go collect the girl's body The old man took the horse and sleigh into the forest and sitting in the very place he had left her He found his daughter alive and well and covered from head to toe in furs A large chest of gold and jewels rested beside her Overjoyed the old man took his daughter and her treasure home The moment the old woman laid eyes on the girl She ordered the old man to take her own daughter into the woods and leave her there The old man did as he was told and left his stepdaughter in the same place He had dropped off his own daughter the night before The girl sat there shivering with cold Soon father frost came the branches and twigs crackling snapping as he moved above her Are you warm enough my lass? Of course not even a crotchety old man like you can see I am not warm leave me alone Logan drummed his fingers against his leg while listening to the story That girl was TSTL too stupid to live so predictable Grandmother continued Father frost moved closer still the branches snapping and popping in his wake Are you warm now my lass? he asked again No, how could I be when you're trying to freeze me you evil man go away Mother frost was so angry he used his frosty breath to freeze her solid The next morning the old woman ordered the old man to go fetch her daughter Along with the furs and jewels she expected to come with Not long after he had left the old man returned The old woman rushed outside to greet her daughter only to find that she was dead And frozen solid She began to scream with the old man So he loaded his daughter and her treasure into the sleigh They sold some of the jewels and bought a new house in the village The girl married a wealthy merchant and lived happily ever after All right boys mom said it's time for you to hit the sack Santa will pass by our house if you're not asleep She gave them each a hug and dad ruffled Logan's hair which he hated He didn't know why his parents insisted on pretending that they still believed in Santa Claus Well, that's not very nice at all is it? Even Tanner thought that I wasn't real Well, they headed upstairs and got ready for bed Babushka had taken over his room So mom had to set up the air mattress for him on the floor of Tanner's room He bounced on the mattress and then slid under the covers and closed his eyes Willing himself to go to sleep so morning and his new gaming computer would arrive faster Logan What do you think father Frost exists? Is Tanner for real? I don't know maybe we should sneak out into the woods and see if we can find him Well, that'll at least kill some time Beyond annoyed Logan threw back the covers fine. Let's go They slipped down the stairs to the dark mudroom and put their coats and boots on over their pajamas Tanner headed out the door But Logan stopped to grab a length of rope from the garage before following The frigid wisconsin wind knocked him in the face as he trudged after Tanner into the woods behind the house It was time for Tanner to grow up It's cold enough for father frost to be here Tanner said, where do you think he'd hide? Logan waved toward the trees deeper in the forest They slogged on his fingertips and toes growing cold despite his warm gloves and boots I think this is far enough. Why don't you climb up that tree and look for father frost? Tanner did as he suggested Seriously, the kid needed a reality check Nope, there's nothing up here. I'm shocked Logan mumbled. What? Calm down Frozen branches snapped and popped under Tanner's weight as he climbed down Should we head back?  Panic raised in his voice to a high pitched yelp as he pulled against the rope I'll come back for you in an hour Logan turned to face his brother. This was your idea.  Logan moved toward the house After dumping his boots and coat on the floor of the mud room. He crept upstairs Logan wanted to give Tanner a dose of reality, but he didn't want to hurt him In case he fell asleep He set the alarm to go off in half an hour Then flopped on to the air mattress and opened his gamer's cosmos catalog He hoped his parents had bought him the computer with the most memory The page started to blur and he decided it wouldn't hurt to close his eyes for a few minutes Logan's heart thundered for a moment. He'd forgotten where he was Tanner's room The beeping alarm had woken him up.  He noticed large footprints in the snow Fresh footprints picking up his pace. He jogged toward the tree and stopped in his tracks Tanner sat in a chair at the base of the tree wrapped in a fluffy down comforter Smiling widely Logan thought their frost is real. Look, he gave me this comforter to keep me warm until you come back and all this stuff Incredulous, Logan looked into the wooden chest It held a brand new computer gaming system several games and a bunch of hundred dollar bills Putting his hands on his hips. He glanced around before saying You could come out mom and dad.  No, I swear mom and dad weren't here father frost is real Sure, he is Logan stuffed his hands in his pockets and debated what to do Go back to the house. I'm going to stay here Tanner moved toward the house lugging his chest of loot Logan dropped into the chair and waited Despite his warm boots his toes tingled from the cold. It wouldn't be long before they were numb Come on mom and dad. It's cold out here and I'm tired No response. He tried again. It's not like I was going to let Tanner freeze I set the alarm and came back for him A frigid blast of wind swept over him and the branches above snapped and cracked as if someone was climbing downward Who's there? He yelled. Are you warm my lad a sinister voice asked Logan's heart jumped into overdrive. He took a deep breath and slowed his breathing Mom and dad must have asked one of the neighbors to scare him Go away. This isn't funny. He said The branches popped and cracked again and a ghoulish old man with long gray hair slid down the trunk landing before him Biting back a screen. He pulled his feet up onto the chair so cold now. He couldn't move his fingers or toes Mom dad he called looking around frantically Are you warm now my lad the man hissed? No, it's it's bitter cold out here get away from me The man opened his mouth wide sucked in a deep breath and blew frigid air over him Logan no longer felt the frantic beat of his heart or the cold he couldn't feel anything Couldn't even blink or shift his eyes. He sat there frozen solid melded to the chair waiting for someone to eventually find him As the sun rose it shone directly into his eyes, but he couldn't close them No warmth seeped into him from the bright sunlight Finally he spotted his brother racing toward him mom and dad following closely behind tears streamed down tanner's face Would logan thaw if they moved him inside would they realize he was still alive or would they think he was dead and bury him? A scream echoed inside his skull But he couldn't make a sound a terrible terrible ending for the young boy Perhaps that's because he refused to believe that I exist or that father frost exists. You'd never know Best to continue believing just to be safe Did you like the stories I told boys and girls? If so tell your friends and family members about spooky santa Do me that favor and that way they can listen to the podcast too And remember you can write your own scary story and email it to me at letters at spooky santa dot com If you want to learn more about the stories that I've told or the authors who wrote them You can find links in this episode's show notes Spooky santa is a registered trademark of marlar house productions copyright marlar house productions Now be a good little boy or girl and join me next time for more creepy tales from spooky santa


Hello and welcome to the session. Let's work out the following problem. It says find the area bounded by the circle x square plus y square is equal to 16 and the line y is equal to x in the first quadrant. So let's go move on to the solution. We have to find the area bounded by the circle x square plus y square is equal to 16 and the line y is equal to x. The circle is x square plus y square 16 is 4 square. So the radius of the circle is 4 centimeter. And we have to find the area bounded by the circle and the line in the first quadrant. So we have to find this area and to find this area we need to divide this region into two regions. One is the triangular region and one is this region. Now to find this complete area we first need to find the point of intersection of the line and the circle. So now y is equal to x is the line which cuts the circle. So we'll put y is equal to x here. So we have x square plus x square is equal to 16. That is 2x square is equal to 16 that is x square is equal to 8 that is x is equal to 2 root 2. Now since x is equal to 2 root 2 and y is equal to x so this implies y is also equal to 2 root 2. So now the point of intersection of circle and the line y is equal to x is 2 root 2, 2 root 2. Now the required area is given by integral. Now as I said earlier the required area is sum of this triangular region and this region. Now area of this triangular region is given by integral 0 to 2 root 2. Since this point of intersection is 2 root 2, 2 root 2. So this line is x is equal to 2 root 2. There is integral 0 to 2 root 2 x dx that is the line y is equal to x plus integral 2 root 2 to 4 of the circle which is given by y is equal to 4 square minus x square. This is integral 2 root 2 to 4 of under the root 4 square minus x square dx. Since in this region x is going from 2 root 2 to 4 as we know that the area of radius of this circle is 4 centimeters so this point is 40. Now the integral of x with respect to x is x square by 2 and here the lower limit is 0 and the upper limit is 2 root 2 plus. Now here we will use the formula of integral of under the root a square minus x square which is given by x by 2 into under the root a square minus x square plus a square by 2 sin inverse x by a. Now here a is 4 so we have x by 2 into under the root 4 square minus x square plus a square by 2 that is 4 square by 2 sin inverse x upon a and here the lower limit is 2 root 2 and the upper limit is 4. Now here we will first put x is equal to 2 root 2 so it is 2 root 2 square by 2 minus now we will put x is equal to 0 so it is 0 square by 2 plus here we will put x is equal to 4 first. So we have 4 by 2 into under the root 4 square minus 4 square plus 4 square by 2 is 8 sin inverse 4 by 4 minus now we will put x is equal to 2 root 2 so we have 2 root 2 by 2 under the root 4 square minus 2 root 2 square that is 8. 2 root 2 square is 8 plus again is 4 square by 2 is 8 sin inverse 2 root 2 by 4. Now 2 root 2 square is 8 8 by 2 is 4 plus 4 plus 2 into 0 plus 8 into sin inverse 1 sin inverse 1 is pi by 2 minus root 2 into under the root 4 square minus 8 is 8 4 square is 16 16 minus 8 is 8. Plus 8 into sin inverse root 2 by 2 plus 4 plus 4 pi minus root 2 into under the root 8 is 2 root 2 plus 8 sin inverse root 2 by 2 is 1 by root 2 4 plus 4 pi. minus root 2 into root 2 is 2 2 into 2 is 4 plus 8 sin inverse 1 by root 2 is pi by 4. So this is equal to 4 plus 4 pi minus 4 minus 2 pi plus 4 gets cancelled with minus 4 and this is 4 pi minus 2 pi is 2 pi. Hence the required area is equal to 2 pi square units. So this completes the question and the session by for now take care have a good day.


Welcome back to prom news Yeah, we saw that giveaway going on for Super Mario 3d world plus Bowser's free re giving away two copies check through the description and The pin comment down below also. Hey look yeah I asked you guys to drop likes on the last video trying to make it the most liked video on the channel in 2021 the problem the most liked video in 2021 is this and Yeah, we got to pass 1.2 thousand likes. That's not something to scoff at but maybe we can come together as a community and pull it off Let's get into the news because we have five big stories here for you guys Not a ton of Nintendo stuff here, but the first couple stories for you Nintendo fans might be of interest after that We got some big announcements, including some PC gaming stuff some Sony stuff Let's just jump into it I'm gonna need my notes yet again for this first story because we're talking sales numbers in Japan So Japan sales are in for week four of 2021 They get come from Famitsu, and we have the Nintendo switch at the number one hardware spot. No surprise there selling 114,170 units the PlayStation 5 also saw a bump over last week selling 25,928 units at the number two spot number three is at PlayStation 4 selling 3873 units and even the Xbox series which includes the X and the S sold 1099 which is actually over a 100% increase week over week The 3ds is still kind of tape rate in there with whatever units are left on the market at 588 now the number one selling software in Japan for last week was Momotaro still at 79,362 number two is Ring Fit Adventure, which saw a 15% increase Over week over week at forty two thousand one hundred and twenty four We have animal crossing new horizons at number three at twenty five thousand one hundred and fifty one We have the soggy a six defiant of destiny the only new release in the top ten software sales For Nintendo switch selling twenty three thousand five hundred and fifty one units Mario Kart 8 deluxe chimed in at number five Seventeen thousand five hundred and three units, and here's actually the rest of the top ten I'm not gonna go over all the games Switch is dominating at nine to ten spots But notably there is a PlayStation 4 game that finally snuck into the top ten for the first time in like two months And that is the soggy a six defiant of destiny. So the PlayStation 4 version of that It's sold less than the opposite of the switch version at fifteen thousand seven hundred sixty one fell at the number six spot But hey, it's proof that you know when games come out on Sony's platforms They could still sell a little bit anyways, although I'm at a point now that I don't fundamentally understand Why Japanese game developers Square Enix and others out there aren't more focused on bringing games to switch when Switch makes up now software and hardware wise combine sales like 99% of the Japanese market. I Mean maybe these companies are just abandoning the Japanese market the way Sony seems to have done, but I Don't know. I just find it strange. Whatever. Let's get into our next story The outer worlds is finally getting DLC released on Nintendo switch the DLC pack we're getting is Peril of Gorgon and this actually originally released last year on other platforms Back in September It costs 1499 back then they have no price currently listed for what the switch one's going to be although we have a release date It's coming next week on February 10th So they don't have a price I know it's with that release dates kind of weird But I'm presuming just like every other switch game that's gonna cost what it used to cost on the older platforms now the thing about the outer worlds is it launched to some controversy on switch in that it was Really really blurry like beyond what we have become accustomed to with diamond network resolutions on switch and It also performed poorly so you would get a super blurry game that had really low frame rate It was just a really bad port However, they didn't release patch 1.2 back in October of last year and well at that point people had kind of given up on the game It actually fixed a lot of this it made the game much prettier and it also Hey fixed the frame rate issue So actually today the outer worlds is actually a pretty good game on switch Perfectly playable perfectly acceptable if you're interested in getting it It is still technically a little bit more expensive than the other platforms But the prices obviously come down from launch because remember it was a 59 99 Launch game that premium triple-a pricing while it is down to 29 99 over on Amazon I'll put an affiliate link down in the description if you're interested in picking it up So yeah, again, it is only 20 bucks on other platforms But if you only own switch or you prefer to have it on the go It's actually a worthwhile play and I can suggest it if you were interested in this game before we're waiting to see Hey, can they actually fix it? Well, they did and they're bringing the DLC now to this next story Move has me tingling a little bit because for those that don't know I am a massive Total war PC gaming fan total war is a series by Sega made by creative assembly and My gosh, it is one of the best like strategy battle games to exist in PC history Like right up there with the age of empires of the world except much more successful over a longer period of time I was there at the very beginning when it was just called shogun total war Total war wasn't even like the overall name of the series. It was called shogun total war We've had shogun total war to medieval total war. There's been a Troy total war. There's been a Rome total war There's been a total war empires There's been a whole bunch of total war games and I pretty much loved all of them Although I haven't played as much of them lately. Well a brand new total war game was announced and of course We're talking about total war war hammer 3 So right now you are seeing the cinematic trailer Released to announce this game and typically I don't like cinematic trailers. However The total war war hammer franchise which has two games in it is actually among the highest rated games in the total war franchise And is obviously established a reputation that you can trust creative assembly to not mess up a war hammer version of total war So yeah, it kind of sucks that we're not going to see some gameplay here But honestly the gameplay is gonna look a lot like the last two war hammer versions Warhammer is a massive IP with lots of books and stories I actually played the MMO when it came out back in the day I don't know a ton of lore behind war hammer So I don't know how exciting we should be over some of these details here But it seems that war hammer fans are pretty hyped for this game One is it's going to be a really really good game and two because of the story direction and what's available So let's get into some of the nitty-gritty details So the product so it revolves around the forces of kids love and cafe with appearances by corn Slammish and tis inch. I don't I'm sure I butchered those names It's set around the reign of Karl Franz and there's actually gonna be nine playable lords I will get a late 2021 release So this is kind of one of those candidates that hey could be delayed to 2022, but whatever Interestingly, they're actually releasing a free patch after the game comes out that if you own total war war hammer one And total war war hammer two along with total war war hammer three It'll actually combine all of the maps into one massive map for the totality of the war hammer franchise See again, I think this is really wholly interesting I'm really glad they're doing this for war hammer fans great for me I just a total war fan like total war war hammer even without knowing any war hammer lore really I actually have one war hammer a book on my shelf, but I still haven't read it I've had it for a long time Yeah, like I don't know anything about the lore and I just still don't care because I know It's gonna be really really good because it's a total war game. So PC gaming news, baby That's what we're talking about. So our next story is about EA in particular battlefield because Andrew Wilson They recently had a earnings call talk about all the latest financials and in the Q&A section He was asked about battlefield and he said hey, that's what we're getting a Battlefield game this year for this holiday. He guaranteed and promised it And he said it's gonna be like one of those defining Battlefield games, of course, they say this all the time and battlefield has been hit and miss over the years There have been some good ones that have been some bad ones The last one was kind of torn in the middle where some people loved it Some people hated it, but it is gonna be the first one built for next gen including xbox series X PlayStation 5 Obviously PC probably expected to also be Cross gen so it'll probably still be on PlayStation 4 and Xbox one But it is the first next gen supposed to be definitive battlefield game We'll see they use a lot of this marketing speak all the time to puff up their games So who knows how much you can believe especially when it comes out of Andrew Wilson's mouth But yeah, he also a note for like Star Wars fans. He said hey, we're not done making Star Wars games either So yeah, they don't have exclusivity anymore, but they're still gonna be making probably another battlefield game I'm sorry another battle front game. And who knows what else is in the pipeline now If you remember yesterday, we actually talked about Sony's financials one of the first times We've actually deep dove into Sony's financials on this channel And we have kind of a follow-up to that because Sony put out there in the Q&A section of their own Meeting that they actually have a forecast for the first full Fiscal year and they call it fiscal year two for PlayStation 5 because it is technically the start of a new fiscal year Which will run? What is it April? 2021 through March of 2022 they run on the same fiscal year as Nintendo and they are saying that hey look we plan to ship and sell 14.8 million PlayStation 5 now that is notably obviously behind Nintendo's pace, which at this point They might hit 30 million this by the end of this fiscal year, which ends March 31st of 2021 But the point is that it is actually a significant bump over what they did with the PlayStation 4 Which was closer to only 10 million units in its second fiscal year on the market So it's actually an increase however as Enthusiastic as obviously Sony is about the PlayStation 5 platform and the demand They did mention that you know what there is actually a Semiconductor shortage in the industry if you ever wonder why it's really hard to get Xbox Series X's and Hard to get PlayStation 5's right now why it's hard to get Certain GPUs out there be it you know the 3000 series of graphic cards or the 6000 series of AMD or even certain processors You know some of the newer processors especially that released late in 2021 like why is it so hard to get these products? Well, it's because there is a massive semiconductor shortage and we need semiconductors to make these products So with a worldwide shortage on semiconductors It's gonna lead to Sony potentially not being able to hit this 14.8 million mark and they admit it Hey, we don't know we could be well short of 14.8 million because they can't make enough Because there's not the available Technologies of the available supply of those semiconductors to pull it off and AMD has already announced Hey with our GPUs and our Ryzen You know CPUs that we are not going to you know catch up the demand for at least six months If not longer and that's just talking about the PC side of things for those who don't know You've been living under a rock But the Xbox series X and S and the PlayStation 5 are using AMD technologies including the new Ryzen and new Ryzen GPUs and all that so Ryzen GPU AMD GPUs Ryzen CPUs, but it's all it's all under the AMD umbrella So yeah, there's a big shortage. So It just might be really hard to get a PlayStation 5 Even in the next year I have a feeling if it could go all the way in the 2022 before we start to feel confident in our ability to buy technology pieces again likely in a post-pandemic world Hopefully at that point. I'm hoping by that point almost everyone that you know is willing to get vaccinated gets vaccinated But anyways, that's it for prime news today I want to thank you so much for tuning in I know not a ton of Nintendo news also kind of a small note I'm wearing the Sega shirt today kind of a celebration of Sega Prime being announced yesterday There's actually a Sega LEGO set out there that's now been approved It was a fan-driven campaign and now like hey Sega's like whoa, we're gonna or like I was like Hey, we're actually gonna get a hold of Sega and try to make this thing reality So I just had to throw on my little Sega shirt to be like yeah, you know what shout out to Sonic my boy, baby My boy shout out to Sega look at all those systems look it all in and by the way This is like a shirt from last time e3 existed. This actually came directly from Sega after I played Mario and Olympic The Mario Olympic games Tokyo games or whatever that we still haven't had the actual Olympics happen yet back in 2019 Anyways, that's what I got for you guys. Thank you so much for tuning in and as always I'm gonna dance my butt out of here


This is a message of solidarity to the Yarlswood hunger strikers. I was there a few weeks ago and I heard your stories and I raised it in Parliament the following week. I'm going to meet with Amber Rudd, the Home Secretary to raise some of the issues you raised with me and I just want to say that you have supporters out here. We believe that the way Yarlswood is run is quite wrong.  The British state has a duty of care towards you and I'm going to see that it actually fulfills this duty of care rather than dealing with you in this dehumanising way. I've also been struck by the punitive deportations. The Home Office has been trying to push through recently and I'm opposed to those and I've been making my feelings public.