Hey, what's up folks? Hey, welcome to the Adafruit show and tell this is the best place to show off your awesome DIY electronics 3d printing or Arts crafts kids projects tour of your makerspace Everything's cool to show off here We're gonna take about 30 minutes to check out everybody's awesome projects You guys been working on I'm gonna start off with some sweet awesome DT key guests here then move on to some of the Adafruit folks and Everybody else in the community if you want to join the link is on Discord.gg slash Adafruit So let's go ahead and start off with tomorrow lab. We got Pippin and Ted. Hello. Welcome to show. Hey guys Everybody yeah, so you guys are plugging this really cool new series that you guys Partnered up with Digi key. It's all the first two episodes. They're so cool It's like doing like design iterating and like three printing some awesome projects you guys want to fill us in Yeah, man, thanks for introducing us And so the show is called potentially genius and it's sponsored by Digi key We bring a guest who comes on and introduces a potentially genius idea And then we try to turn it into a potentially genius thing in 16 hours Using all the tools and skills at tomorrow lab. Sometimes it takes a little more time, but the results have been super fun And there's new episodes monthly on Wednesdays in the middle of the month Nice Including today Today just launched. Yeah, really cool like a bottle cap sorter for doing like recycling Exactly. Yeah an episode two we worked with Jasmine who's a social entrepreneur and former UN fellow now based at the in Phoenix and She's creating the small-scale operations at her university and also in her home country of Ghana to expand recycling and improve recycling outcomes So it just launched today. We use a cool like color sensor and little solenoid from a different Digi key And then yeah an episode one which launched last week We teamed up with a filmmaker and skate enthusiast named Sean from Brooklyn and we created this in line electric inline Skatebreak so it just pops into the skate frame replacing one of the wheels and kind of gives you this variable braking resistance with the rear wheel Awesome. Yeah, I was skates. Yeah, it was so cool Do you guys like release like the designs or like any of the like schematics for building your own or it's just like to show off like what the iteration process is It's more to show off the process. Definitely. Okay. Yeah, we're big open-source fans and So, you know audience is kind of welcome to run with what we've created But we're you know depending on the project We'll kind of continue the development thereafter. In fact, like we've been doing some more testing with the skate break Which you know, we can share some test footage if you guys want Exclusive clip guys. This is very exclusive So this is a little demo of how it works. There's a there's a remote control that's off camera that's actuating this solenoid or sorry the servo to move the motor You can see it adds either some resistance or a lot of resistance or no resistance to the rear wheel So it gives you this This brake when you need it and and it's gone when you don't And there's three levels of resistance off half and full and we had this fun led light on the back to kind of communicate those modes and You know flaws of prototyping are real even for us. So the remote had had some short range But you know, it's still a good proof of concept. So here just two quick tests of Going downhill first with it off And then again with it on works Works you can sort of see this like ghosted comparison here. Yeah, it was cool. It was a good idea to do the comparison that way Yeah Nice And there's there's one more test going down a steeper hill Um Don't turn into a ghost use chronic So, you know for a lot of people like in line skating can be daunting trying to trying to start it Particularly around the speed that that you can you can gain Um, so yeah, we think this could be something that can get beginners more on board Or also if your professional could add more resistance to your your training regimen, for example That's just another another example I was gonna say or as you get older and you get more scared like oh no, am I gonna Bus something and my kids gonna be like Because I was telling you guys I like do a little the way I stopped But I would do like a little curve like swirl and then do like the stop that way But now I'm like kind of scared to do that Well, this is for you Yes, definitely. Yeah, so if you guys do end up like releasing some of the files Yeah, I'll definitely try it out But uh really good inspiration for doing like at least like a break light on the back definitely interested in uh You doing that like the way that it was connected to the back So super awesome. And yeah, everybody go uh subscribe to the digi-key channel There's a think can can you fell us in with the next episode? It's gonna be It's only in production right now, right? Is that ice cream? I think that's the note That's the airline sensor. Oh, it's airlines. Um, so we uh, yeah Also Ice cream is coming. It's just in time for winter. We've made it ice I won't spill all the beans but it's an ice cream related project And sometimes we have challenges with ice cream cones. We solved those challenges. Oh, okay Some ideas out Yeah, and the one next month is is a little bit um, it it's more of a serious issue of these uh These like um fume events that happen on airplanes sometimes We're like the jet fuel gets sucked into the cabin So we created kind of like a safety sensor for people to see what's going on And so uh, yeah a lot of different we tack a lot of different like topics and um a lot of different sensors And it'll be um, it'll be fun. We're shooting our sixth one right now By the way, which we don't know the topic. So we're waiting for people to to give us ideas For the topic should be I'm sure that's you in discord, right? Yeah Yeah, awesome. If you don't get it to it today and you have an idea post it anywhere on any of the videos We read the comments. Um, I know all of the comments Cool. So yeah, thanks so much for sharing guys. I'll I'll be waiting for this one. Thanks a lot for sharing Thank you. You know Pedro Wait, wait All right, let's check next time we're gonna check in with some folks for made of fruit We got melissa up and she's got some new stuff to share. Hey melissa. Hey, hi, you want to show my screen? Yes, you got it This is a circuit python coder that i've been working on here and this is just running locally on my computer But i'm gonna go ahead and hit connect And then once that does i'll bond it So one thing i've been working on here is adding, uh, some of the file functionality here So i have it right now. So you can kind of go and see what's on the board And then like go into the library Go here. So i have file navigate some basic file navigation built into it So i want to do it so you can say choose where you want to save a file where you want to open it and such and so uh, it doesn't actually open the file yet, but Um, coming to the surface on it. Yeah, i love the UI. I like the icons Awesome. Thanks. Yeah, they're just uh font awesome Icons here. So, okay. Oh, they fit perfect. Yeah That's great So coming soon. I can't wait to try it out. Hope to see more next week. Yeah. Yeah All right melissa. It's great seeing you I'll catch you next time. Yep. Thank you All right. Next up. We're gonna check in with jeff and jeff screen We can't So, um, I'm working on a new camera module. Actually, I was working on it last week You might have seen a one minute video of it, but we made it go a lot faster. So, uh, this is this is like, uh my ceiling everyone's sick of seeing my ceiling but This used to be running at like two frames per second And we found a couple of simple changes and now it's running at 16 so Really great for like a preview uh mode for a digital camera We're also saving jpegs off it, which is neat. Um, I need to figure out how to show a demo of that There are all sorts of effects modes that this can do like a sepia negative black and white There's this is the best sensor yet. We really like it and we're just working on making a great driver for it How? Yeah, so this is the ov 56 40 And the code is up on github Um, it is in the circuit python bundle and it runs with circuit python 7 And the best choice is the esp 32 s 2 microcontroller Anyway, so look for more stuff related to that kind of trickling out as time goes by Very cool. Is this the one that just got added to the the shop? The um I think they added the the open mv board. Um, that's kind of its own thing But they've got a fork of micro python that is for computer vision stuff Which is really cool and if you're into that you should check it out because You get all the micro python and you get some really nice Kind of cherry-picked Uh machine vision type stuff. So it's a neat product, but this is different than that. So Okay, okay. All right. Thanks for your fun. Oh cool. All right, jeff. Great stuff. Yeah, have a good one. Thanks for having me Right Okay, next up we have jp john park Hello, it's me and look. There's my demo right there. That's perfect. So, uh, what's this error? It's getting I don't know what that's about. I'm gonna close that.  But the answer is no. No, we're not done So this one Uh is hand wired and so you can kind of see this comes with like a little acrylic back on there You can see there's just four. Uh, I used the kale box navies, which are super thick and clicky And then I've just uh soldered the uh a single wire for ground to them And then the other four pins are going to four GPIOs on there and then I've screwed the little case bottom on plugged it in and This was partly a response to the stack overflow copy paste Macro pad which I think is three keys and I'm assuming they're actually Intending the first button to actually be your control key or your command key and then copy paste I decided to do this one as four separate functions that are They each of them contain their There what's the word I'm looking for like multiple key presses the that are necessary. So this one does Oops, I'm not clicked in here. Let's get in there. Uh, this first button does command a for select all or control a And then I can cut that with the next one. I can Paste I can select all and copy I can paste paste paste to the end of time So, uh, I thought that was fun to have those functions on just a single Uh little set of buttons. You can of course reprogram that to be anything you want Uh, and one of the nice things with with this is in our usb hid stuff in circuit python You can actually be somewhat, uh operating system agnostic for certain types of codes. This one actually uses the Keycode dot gooey And then keycode dot a for all Or c for copy. So the word gooey actually is a stand-in for command on mac and control on windows at linux So there's some in hid that are generic enough that you can then go and plug this into different Different machines, which is kind of fun. Um, super useful So that is a guide that I just wrote up and it's going to be coming out soon I don't think it's published yet, but it should be out soon pretty simple guide pretty straightforward But a fun little sort of free wiring soldering project. No pcb required And then I will tease one thing which is for tomorrow I'm going to be working on a little halloween thing Which is going to be a silhouette for your window Which will be a little cat cut out with with backlighting and on the front of it We've got a little cute little eyes there with our led glasses and some code that I found on discord on our discord There's a username squid.jpeg who wins for best username ever and they posted their code on github So if you go to our show and tell channel on our discord You'll find more about it and there's a few different versions including one that uses a nunchuck We nunchuck to control the eyes, but I made a little modification to make these eyes a little taller and slittier like a cat eye But that's that's a fun way to add some Holiday stuff to your window as halloween approaches. So stop on by the workshop tomorrow and I'll be showing that and more Awesome. Thanks. Thanks john. Thanks john Thanks guys. See you late Okay, next up we'll uh check in with rj Oh Hey It's good It's cool Yeah It's just a whole bunch of uh led rings this leds and of course my favorite a different trinket with a small power Very simple very fun build but something Yeah, I saw the uh, right if you did on digikey Yeah, that was for a different project Okay. Oh, oh no, that was the shoulder bot. Oh, okay. Yeah, this is a mass thing. I did on my own time Oh, okay. Okay. Awesome.  Definitely. Um, maybe it's just I'm still I'm working on making another one to see if I know what I'm doing because I have a personal rule of make it once it was luck make it twice No, you're doing that's a good rule I can't make it twice then I know there's something wrong. It shouldn't be there Oh, oh no What was the process like for getting like the uh, like to fit your face? Did you have some measurements or like, um, I just took some notes for some of the other cosplayers to do Iron Man suits and they usually size it up A certain level depending on your head and one person did Iron Man helmet He had to size up on his parts to 970 So I just tried that and it works and I was like, all right, cool now I have a percent of the one sign slides everything up Nice It's my first it's my first wearable mask. So Nice, it looks good. Thanks I'm gonna guess you're gonna incorporate it with something from like squid game or something, right? That's the trendy thing No I know I've watched all the spoilers and like behind the scenes and they did the CGI he keeps egging me to watch You know You don't do it don't fall into peer pressure Yeah, hopefully I'm going to I want to go the second one. Hopefully tonight because it shouldn't take me too long to build another app on the parts and Depending on how it comes out. I'll post about it Next up we're gonna check in with Michael give a second here, Michael. Hello. Hi. How are you? How are you? Good. Um, so last week I got this awesome board, which is the glasses driver board and I Actually ordered two of them and I actually made A watch with it. Um, which I've made a watch before with it. Uh, this is technically version four now So let me switch my camera over Um, and I'm using uh the Adafruit 128 by 128 OLED Um, this is version two, which was like my last one I made last year that I've been like wearing over the past year Uh, and this one was cool, but It was multiple parts and things like fell off and you can see I had a super glue it over the year So I want to like redesign it. Um To be like one solid thing. So like when I smash it into something it's not going to like fall apart Um, and the way I'm able to actually get the neopixels working, uh, which the neopixels are supposed to emulate like actual hands on a like traditional clock. So red would be our blue would be a minute. Um, I actually Connected it to where the button is There and Basically had it which this is like known as pin four in the software So I had it kind of hacked it. So it will Be able to drive the neopixels Uh, which is kind of fun I think that it's like the um Is there a name for the apple watch band? Is that what that is? Oh, no, this is just a I made it so it's capable with It's kind of like a clone of the apple watch band But any 22 millimeter band should work. So now that's a really good idea how to be compatible with the standard bands Awesome. All right. Well, thanks for sharing Michael. Yeah, that's what I love the look of it Thank you. Great. Have a good one All right. Uh, next up we're going to check in with Liz Let's say DIY. Hey, Liz. Hey, how's it going? Um, so last week I started talking about how I was doing kind of a disconnected, um, co2 sensor Uh, so I had signed up a case. Uh, so this is the first iteration. Uh, it's a little too snug I tried to get it like really small. Um, but it's not quite large enough. Um, but luckily, uh, I designed a confusion so I can just change the values and make it a little bigger Um, but basically, uh, I've got a pcb mount. So the feather mounts on the bottom and then the stemma sensor on the top and then inside, uh, this would be the power switch, uh, and then Slot for the charging port. This does fit the, um, the pink Uh, usb c cable really nicely so that that's a victory at least. Uh, so just some resizing and it should be in business and this is the cdc 41 sometimes I don't know the uh, the air quality sensor at cd 40 sometimes the new one, right? Yeah so a little like Like on the top of So sweet. Let's see in the progress. Good work. Thanks. Sure. Thank you. Hey All right, we'll check in with you. Uh, next time. Thanks, Liz How's it going? All right, next up we're going to check in with Andrew s Hello, Andrew How's it going? Good. How are you? Good. Thanks. I'm gonna show off my project here. I've encased my macro pad in solid wood so this is a giant chunk of tiger wood And I've cut it like a band saw box so along this edge carved it out inserted the macro pad and Right now it's running the macros from the hotkeys demo But the one thing that I was doing is um, it's really nice. It's just a luxury item And I love I got the little squishy silicon on the bottom So that keeps it nice and it's heavy Um, the one thing I did do is this uh wallet sized Um, for credit card sized magnifying glass. I got a bunch of them cheap And it's hard to really demonstrate but uh It's got this kind of weird effect because it's a Fresnel lens And it shows kind of a bubble here. So Anyway, that's uh Oops, there's one on the screen I'll show you I just got the keys doing a bunch of different colors and stuff right now These are three of my forever keys and then I'm going to replace these eventually but Yeah, it was a fun project. It took me a long time and I don't want to make another one really But I might make a mouse and maybe one of those uh jog dials from Todd bot. Yeah Make a matching set so I can put a mouse in the jog dial But anyway, it's just lovely wood. Yeah, how did you hollow it out like with the um Sorry the bag with like the the wood um tool names is is it a so a bandsaw bandsaw there came through here chopped off this face plate and then Which you can't see well my bad glue job. You can kind of see it, but I broke the edge right there and um Cut through the face and then I cut the face out and then uh Separately hollowed out this thing with a router So it's deep enough and I put a one inch foam in there. So it's it's nice and Tight in there and I've got uh Just enough tolerance is those uh little silicon usb things too were nice to sit in here While I was doing all the finishing and the sanding it was uh good protective Otherwise, it's uh kind of just wrapped in painters tape in there, but it's permanent And my one my one flaw is I forgot to put the the reset hole So Luckily, I upgraded to uh circuit python 7 already. So I don't need to do anything right away Yeah, I've been working Your keys were great I thought that might be a little too much wood, but I was thinking if the ones that you milled would be nice If I did it in a similar fashion Yeah, you can never have too much wood So again, this is really fun and I I don't know if you can quite get the effect of that Fresnel lens, but it's uh In person a little bit. Yeah, that's pretty cool That's a good idea. It's a good touch late at night too. The light kind of bounces around inside there as well And you can see it on the inside Yeah, one project so much Andrew. No problem. Thank you. All right. Thank you so much. All right, next step for me to check in with James Hello, James. How are you gentlemen? Hey, how are you? Over the weekend I wrote some software and uh, it's interesting to Not to have to copy circuit python files back and forth using like windows explorer or whatnot And I wanted to find a way to to make it a little bit easier more streamlined I don't know about everybody else. I got a folder full of circuit python files So I wrote this quick little thing where you open up a folder any folder that you like and you can go and change What circuit python file that you're working with? and ultimately I now have a you know, like a one-click solution to be able to change anything that's happening on in circuit playground express I have here and in addition to that I also wrote an android version which uh Easy enough. It uses a web sockets to connect to the windows software Okay, I hope if I actually bring this down a little bit Try to get it all in the scene at once. Yeah And uh, it goes and grabs the list off of the windows pc off the folder I told it to and I can tell it remotely to uh to do the same exact stuff simply copies the code That py file over and uh And makes it do stuff for me said having to play with windows explorer and whatnot Oh, that was annoying Uh, and some people showed some interest in the in the forum and discord and I thought I'd show it off Yeah, this is awesome. It's like your own custom gi Basically, yeah Yeah, and also it was kind of fun having to edit the files. I can simply right click this takes like another program It's a patchwork of programs really Um, this now can go and open up with uh Plus plus my favorite editor I can just simply right click on stuff instead of having to go to each and every code dot py And and go right in and edit and then save and and simply goes and copies a file right back over to the microcontroller again Very cool I don't know pretty easy. Yeah, this is great. Thanks for sharing. Yeah, definitely link to this video Anybody's interested. This is awesome. Thanks. I might put it on github. I'm gonna look into it and get an account and stuff first Oh, yeah, yeah Thanks so much. All right, james. Thanks for sharing Hey, all right, and we're gonna finish it off with uh, graham and matt Hello, hey guys to the show Oh, can you guys hear us? Hello? Yes. Hey guys, just fine. Awesome. How are you guys? We are uh, we are part of team life long We are a Kickstarter project that just started off recently Nice, and we are working on this guy Let's show this off I've seen this. Yeah There we go So, oh fun Yeah, yeah, it's uh, we're we're focusing on one-dimensional video gaming and it all This big addressable rgb led strip and inside this controller We have a feather m4 and it's running a fork of circuit python to Uh, run all our code and uh, and yeah, we figured we could give a gameplay demo or answer some questions Yeah, yeah, please share what we're doing So this game is called light tennis. It's like ping pong Or table tennis, but in a light tube. Yeah, except one d It's pretty uh, it's pretty simple, but uh, we spent a lot of time designing it and getting the game feel right and Uh, I don't know. I'm really excited about it If you press it closer to the end it moves faster and pushes this middle blue light towards your opponent So the goal is to time it so that it hits that you're very end This is a perfect party like game like easy cleanup like Exactly exactly and it's like nice and glowy so it goes along with like any like party atmosphere Yeah, this is awesome There's a much more aggressive game that people get a little bit crazy over called tug of war And it's the ultimate button mash battle. Whoever presses faster. Let's go ready. Oh three two one Oh Yes Although I don't advocate this is great for like gambling too like who's gonna win Yeah, and and one of the things that we really want to prioritize is uh, kind of keeping that, um Um Ease of development with circuit python So we want to make it easy for developers and maybe even non-developers to yeah Yeah, I have to create new games and new content for it Helper tools worldwide so anybody anywhere can create games and applications and maybe light visualizer for light pong and then you can download it from a app So very cool Yeah, definitely post the link in the discord. This is awesome. Yeah. Yeah. Um, I'll post the link in the discord Um, we're light pong on kickstarter play light pong dot com. We're hiring right now So if you're oh, but if anybody wants to join the team feel free to reach out. Yeah Yeah, uh, we have a jobs board on it if you dot com slash jobs and you guys can post there. Yeah, okay Good little honeypaw for anybody who's watching Thanks so much I think that's good to do it. Yep. All right, awesome show and tell guys is all right Thanks for ready. Don't go anywhere and just a couple of minutes the marge pill two minutes two minutes that's a couple in it We're uh, we'll see you guys uh next time Yeah, thank you everybody. Bye


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Hi, I'm Zor. Welcome to a new Zor education. I would like to continue solving trigonometric inequalities. They are not very difficult. There are three of them, which I would like to talk about today. So, well, let's just try to do it. I'm sure you tried to solve it themselves first. Do it just by yourself, thinking about these equations. The inequalities, it's very important. So, regardless of the fact whether you have solved them correctly or not, so let's just examine what I consider might be a good solution. All right, the first one. Sine vex plus cosine vex less than square root of 2. Okay. Obviously, we can do it in many different ways. I think what this particular problem actually is trying to make a point, before solving anything straight without any thinking, is to just draw a graph of this function, sine of x, cosine of x, and then together get some curve basically and try to maybe resolve the equation when this is equal to square root of 2. Yes, you can do it this way. But obviously, it would help if some very simple transformation would help you to simplify this particular inequality. So, how to simplify it? Well, in this particular case, I don't know, this square root of 2 actually kind of reminds me something. It reminds me that square root of 2 over 2 is a cosine and a sine of pi over 4. And if I will multiply by square root of 2 over 2 left and right part, it looks like I'm multiplying by sine and cosine, and that would look like a formula for cosine of the difference between x and pi over 4. Right? So, I multiply left and right part by square root of 2 over 2, but since it's the same, so instead of square root of 2 over 2 with the sine of x, I will put sine of pi over 4, and instead of square root of 2 over 2 with a cosine, I will put cosine of pi over 4. And then it will be cosine of x times cosine of pi over 4 plus sine of x and sine of pi over 4, which is the formula for cosine of the difference between x and pi over 4. So, cosine of x minus pi over 4 less than 1, right? Now, cosine is always less than or equal to 1, and at what particular points the cosine is equal to 1, which we have to exclude actually from the solutions. Well, remember the cosine is abscissa, right? So, cosine is abscissa, and the cosine is equal to 0 at the angle equal to 0 plus 2 p m, right? So, x minus pi over 4 should not be equal to 2 pi m, where m is any integer number, because v is 2 pi m is exactly this particular angle, which means that x should not be equal to pi over 4 plus 2 pi m, where m is any integer number and any other x, but this one would satisfy the inequality. That's it. So, again, the trick is to multiply both parts by this expression. Just, you know, think about this. If you have sine and cosine in this particular combination, it might actually be helpful to multiply it by square root of 2 over 2, which gives you a little bit simpler formula, basically. Second one, sine of 2 pi cosine of x greater than 0. Well, this is kind of a complicated thing, so you cannot really, like, immediately come up with the solution, even graphically, quite frankly. I never actually attempted to draw a graph of function like this one. However, we do know how to solve this particular inequality, right? So, we can convert condition on a sine into condition on the y. Well, in this case, y is this. So, when exactly the sine is positive, well, sine is ordinate, right? Remember. So, these angles, they all have positive ordinate. So, it's from 0 to pi plus 2 pi n on both sides. So, these are conditions for the angle sufficient for its sine to be positive. So, this is exactly the condition which we would like to impose on whatever we have under the sine, whatever the argument to a sine is. So, what I want to say is that 2 pi cosine of x should be less than 2 pi n and greater than pi plus 2 pi n, 5 plus 2 pi n. And this is 2 pi n, right? Well, obviously, we should reduce by 2 pi. It's positive, so everything is fine. Over 2 plus n, right? That's what we have. Sorry, that's 1 half. We reduce by 2 pi. We divide it by 2 pi, so it's n. This is the cosine. This is 1 half. And this is n. This is basically a condition equivalent to this one. We did not really lose anything. We did not add anything. So, instead of solving the original equation, we will solve this. And by the way, this should be simultaneous. So, it should be like intersection of two intervals or more intervals with the regions. All right. So, let's talk about this and consider this to be my problem. Now, let's think about it. Cosine is a restricted function. It's from minus 1 to 1. So, n cannot be very big positive or very small negative value because the cosine will be outside of this range. Now, can n be, for instance, minus 2? If this is minus 2, this is minus 1 and a half. Cosine cannot be less than minus 1 and a half, right? Because of this. Now, can n be minus 1? It's minus 1 and this is minus 1 and a half. That's possible. So, one of the things is, where we should look for solution is this. Can n be equal to 0? Well, yes, because it will be from 0 to 1 half, which is also good. From 0 to 1 half, it's possible. Can n be equal to 1? No, because 1 and 1 and a half cosine will not be in between these two values. So, basically, we have reduced our inequality original to two independent solutions. Solutions to this and solution to this. All right? Well, that's kind of easy. So, let's just solve them. For the first one, let's consider the graph. That would be easier. So, this is 0. Pi over 2. This is pi. This is minus pi over 2. And this is minus pi. Now, we also have minus 1 half and 1 half. So, this is 1. 1 and minus 1. 1 half is here. Okay, let's consider the first one. The first one is from minus 1 to minus 1 half. Now, minus 1 to minus 1 half seems to be like this is minus 1 half. So, it's this particular area and this particular area. Right? So, what is the value of X if cosine is equal to minus 1 half? Well, that's pi minus pi over 3, am I right? Now, cosine of pi over 3, which is 60 degrees, is equal to 1 half, right? Cosine of 60 degrees and sine of 30 degrees are 1 half. Now, when I'm changing from pi over 3 to pi minus pi over 3. Now, obviously, if this is pi over 3, this is pi minus pi over 3. And abscissa would be the same absolute value but negative. So, we need minus 1 half. So, this particular point is pi minus pi over 3, which is 2 pi over 3. Now, this point correspondingly is minus 2 pi over 3 because cosine is an even function. And these are obviously minus pi and minus pi. So, the solution to the first one is to make cosine from minus 1 to minus 1 half, we have to make abs from minus pi to minus 2 pi over 3. Looks like I'm right. And from 2 pi over 3 to pi. This is this area. And obviously, I should add 2 pi and 2 both sides. So, these intervals are solutions to the first one. Okay. Now, solutions to the second one from 0 to 1 half. So, from 0 to 1 half. So, it's this. From 0 to 1 half. It's this piece and this piece. The cosine would be from 0 greater than 0 but less than 1 half, right? So, what are these intervals? Well, 0 is obviously minus pi over 2 and minus over 2. And 1 half would be pi over 3 and minus pi over 3, right? Because the cosine of pi over 3 is 1 half. So, in this particular case, we have from minus pi over 2 to pi to minus pi over 3. From minus pi over 2 to minus pi over 3 and from pi over 3 to pi over 2. That's this piece. And again, I should use 2 pi and in both cases. So, these are four different groups of intervals. Now, why groups? Because of this n. n is any integer number. So, for any specific n, we have four intervals. Which is this one, this one, this one, and this one. Where this particular condition is. Where original inequality holds true, right? Okay. That's it. So, what did we do here? We categorized it. We reduced the original quite complicated inequality into a combination of single ones. So, from sine of something, sine of 2 pi cosine x. We have reduced to condition on 2 pi cosine x. And from that, we have reduced the condition on x. So, it's like two steps procedure. And the third problem which I have is cosine of pi xy greater than 0. Okay. Now, the first question you might ask is, hey, we have only one inequality and two different variables. Well, that's right. Which means that solutions are not expressed in some kind of inequality for x. But it should be an inequality on x and y. And how can we express it? I mean, graphically, in the previous cases, we always had some kind of interval. X from this to this or from that to that. That's one dimensional intervals on the line. Well, this obviously is a two dimensional area on the plane, on the coordinate plane. The question is, how does it look? So, my purpose right now is basically express it not only analytically and analytically to express it. It's very easy. The cosine is positive as you understand from here is the graph. So, this is one, this is minus pi over two and this is pi over two. Right? So, cosine goes like this. So, it's positive between minus pi over two and pi over two and plus two pi n are this, right? So, I can very easily express the condition on x and y. Y basically saying that pi x y should be from pi over two plus two pi n to minus pi over two plus two pi n. Right? But what is this? I mean, in case of intervals, you just see, okay, any x which belongs to this interval is a solution. Now, which pairs of x and y belong to this particular solution? Well, for instance, x and y equal to zero does fit, okay, but what else? So, we better express it graphically. So, my point is right now you cannot really express it analytically better than this one. I mean, obviously you can just resolve it for y or for x or whatever. But it does not really give you the feeling of what exactly the solution is. That's why I would like to approach it graphically. All right? So, how can we approach it graphically? I suggest we slightly change this. Well, first of all, we obviously can reduce it by pi. So, I will have x y between minus one half and one half, x y between one half plus n and minus one half plus, no, two n, sorry, two n. Now, for every n, we have its own area of x and y, you see? Let's say for n is equal to zero, we have x y from minus half to half. Now, what is it? Well, let's just think about it. Let's put n is equal to zero first. Now, our x y is from minus one half to one half. Now, how does it look graphically? Let's think about it. Let's build the first one, x y less than one half. Well, let's build x y equals to one half. That probably divides the whole plane into two parts where it's less than one half and where it's equal to one half. And the division point is where it's equal to one, where it's equal, right? So this is hyperbola, right? Because for positive x, we can say y is equal to one half of one x, right? Now, y is equal to one x is just a hyperbola. One half, it's just slightly squeezed down by now. And on the negative side, it's this, right? So these are two pieces of the curve which supposed to divide the whole area into the whole plane into areas where x y greater than one half and less than one half. So in this particular area, let's say take zero, zero. Zero, zero is obviously less than one half. So the whole area which is in between these two is less than one half. Here x y would be greater than one half and here it would go so greater than one half. So that's how we have divided our plane into different areas, right? Now, what about x y greater than minus one half? Again, first of all, let's draw y is equal to minus one half of one x. Now that would be this, right? These two curves represent this particular graph. And now we need again, how does it divide the area? Well, same thing, point zero, zero is greater. So in this particular case, again point zero, zero which is in between these two belongs to the area where x y is greater than one half. And whatever is outside, this is x y would be less than minus one half. This would be x y would be less than minus one half. So what I want to say is that this particular combination of these two inequalities is this diamond kind of shape. Well, I mean it's obviously not a diamond because it's infinite in both cases. In all four sides actually it's infinite. It's asymptotically approaching the axis but it still looks like a diamond basically, right? So that's what it is. This is this particular area which is represented by this particular two inequalities. When x y is greater than minus one half and less than one half. It's this area in between these four lines. Okay, fine. That's done for n is equal to zero. Let's move on. Okay, so far we have this particular piece of the plane as a solution to our initial inequality. Now, let's move on with n. Let's say n is equal to one, all right? So when n is equal to one, I have here two minus, so it's one and a half. So one and one half x y less than x y less than two and one half, right? What is this? Well, again, let's build two curves. First, x y is equal to one and a half and then we will take whatever is greater. Now, what's the difference between this which is actually y is equal to one and a half times one x? What's different from this? This, if you remember, was one half times x, times one over x. And this is one and a half. So it's higher, basically. So if original was a standard hyperbola y is equal to one over x squeezed down by the factor of two, then this one would be stretched out by factor of one and a half. So it would be higher. It's always higher than this one. And what we are looking for is values which are greater than this one, which is this one. Now similarly, with a negative x, it would be this and the area would be this. How about two and a half? It should be less than two and a half. So that would be, the coefficient would be two and a half and would be probably even, I mean, hyperbola even higher than that. But now the condition is less than. So combination of these two would be area in between these two hyperbolas. So this is not a solution. So in addition to this diamond kind of thing, we have a stripe from one and a half xy to two and a half xy. Now obviously if we increase n, we will have, for n is equal to two, we will have what? Three and a half and four and a half, right? So it would be even higher. So again, two stripes will be added. It will be here and here, et cetera. So I will add stripes. Now with a negative n, I will have let's say minus one. If it's minus one, so it would be what? Minus two and a half and this is minus one and a half. Well, obviously it will be stripes on this side. So in addition to this diamond kind of area, which is the result of this thing with n is equal to zero, I will have additional stripes. These stripes would be added as n is equal to one, two, three, four, et cetera, positive. And these stripes would be added as n is equal to minus one, minus two, minus two. So the combination of all of them is this type of a geometrical area, diamond like in the middle, and then stripes around it. But all of these stripes are asymptotically tend to access so they are narrowly, narrowly and narrowly as we go either to the right or to the bottom or to the left or to the up, right? So that's basically how this area which represents the solution to the original inequality looks like. Well, it might be a little complicated, I don't know, but I think this graphical representation really helps you to better feel what exactly all these inequalities are. And don't get scared by the fact that you have two variables and only one inequality. It just means that the result would be a two dimensional area on the plane rather than one dimensional interval or a set of intervals on the line. Well, that's it. I do recommend you to go through these problems again just by yourself. And don't forget that everything is on Unisor.com where the whole educational process actually can be utilized by you. And it has exams for registered students. Your supervisors or your parents can enroll you into different topics, can consider and mark these topics as completed. If the exams which you did, which you took are passed to their satisfaction, etc. So again, try to do it yourself as much as possible all these problems before you listen to lectures like this. And that's it for today. Thank you very much and good luck.