Transcripts
1. Trigonometry for Physics: you are in No. Two last years of high school and you took physics. But it is so hard, it doesn't have to be that way. Physics is actually quite easy. And once you realize this, you will also realize how this subject is fascinating. Many of my students get stuck on. They have trouble applying when they learned math class to physics. This is why I created this course, a bridge, a bridge between math and physics. In this course I review with you all the math. You need to feel comfortable in your study of physics. At high school level, there are three sections. A dribble. Well, you will learn how to rearrange physics equations effectively and reliably vectors so important in physics off which the manipulation you will learn to master. And finally, we will dive into trigonometry and its application to vectors. As about this dispersed in the course. They are also talks of little practical tips to make life as a physics student easier, follow their schools with a pen, paper the calculator and be ready to pull the videos. Lessons are punctuated with solved examples and questions for you to train. They are also videos that compile exercises, including exam questions for you to work and in these videos, a solve these questions in grand detail on a white board. Take this course followed diligently. Work on all the exercises and math will never be a problem again Studying.
2. What is an Angle?: before we dive deep into basic trigonometry, it might be a good idea to define what an angle is. Yeah, What is an angle? Imagine a Euclidean space with Kardashian access. Thanks. Why? Oh, that's a legion. And I'm going to define the point which is at the distance. Oh, from Do you know what is a locus? A locus is a group of points which will share a common property. What would be the locus for which all the points, right? The distance R from Oh, so you could have mentioned in point here, which is a distance. Are another one there, Andrew? One here, etcetera. Filing all the post together. We'll do you get a circle. So Circle is the locus for which all the points said the same property, which is to get the distance off the all we do. Well, I'm looking at the so called the screen. We can t find a point ed here. What is called this distance? This is Gordon Arc, An arc of a circle. Let's name it s It's a distance is described in meters and we have a radius here. But he's also described the meters. Let's consider Oh, and I'm like a vector Easier. You see that if I only in my vector differently well s is going to change. So I have to define the number to show the orientation of the vector. That's cool. It picked up that would represent the orientation of the vector belatedly to the X axis. What I notice is that when Tita goes up, s goes up actually in a proportional way s is proportional to the I could also fix titter an increase of radius. What would happen to us? They imagine the biggest circle well, as with increase. So if all goes up, s goes up and actually it's in the proportional way that gives us a relationship between these quantities. And actually relationship is that oak is equal to the radius multiplied by quantity called an angle, which is a quantity off orientation of the victim. What would be the unit of this? Well, let's put it a subject to Tyco's s is a distance in meters. Oh is also a distance in meters. So we got meters of the meters. So the angle theta has no unit. No, it's a racial distances and Nangle is a ratio off distances. It has no units. So what will be the value of this number when the ark equals radius? If s equals? R. It was one that is what defines the rage in one reagent so graphically you take our and you imagine the same distance on s. It would be something like this by me. And then you have the angle theta here. Well, that's about 60 degrees by the I. I see one region is about 57 degrees. I drew another circle is much nicer. Why did I do this? Because I want to play a little bit. What would be the angle if the ark was equal to the perimeter? Meaning that the ark would go all around the circle? M and N would actually be the same point. What would be the angle? Well, in order to determine this, you know that the angle is the ark divided by the radius. Find this ratio. You find the angle. So what you produced Just take a piece of strength which is the size of the regis, and then count how many times the piece of string fits in the ark and the value you find this to pipe. Ending up with having the arc of a full circle equals two pi r, which is a perimeter. What is he angle in reagents when the ark is a full circle. But we just saw that it's two pi, but in physics and engineering, we use another unit used agrees there are 360 degrees in from So for these, um, agents. Do you How do you convert from one to the other? Well, here are heavily to table which I can use to perform across. For that, I suppose I'd have the angling radios and I wanted to degrees. So this is what I want the other three things I have. So I just need to multiply this one by this one and divide by that Quetta in degrees would be equal to deter in regions but to glide by 3 60 Divide by two pipe. So I really like this to make it move. Agreeable to read. That's to succeed by by two parts 1 80 divided by pi by deter in Wait. What if I want now to find time Iranians from 2 10 degrees, same thing. I just multiply these two and divide by that one, so that would be tied. Reagents equals two pi in Paktika in the breeze divided by 3 60 And if I rewrite it better giving me pie of a 1 80 I titter in the greets. That's how you convert for one to the other. Now you don't have to demonstrate everything. Is that to remember? Just remember 1 18 pie 1 80 of a pie is a bigger than one. So when you have radiance, you know you need to meet a car by this which is bigger than one to get this in degrees because the number corresponding to angles in degrees is much louder to the number corresponding to angles imagines. So you know you need to lose a guy like this one. If you want to go from degrees to radiance, you know the number will be smaller, so you might gather up I over 1 80 which is not less than one. What is the arc length on a circle of radius? Five meters. When the corresponding angle is 100 degrees, the radius is five meters. The angle is 100 degrees said look straight forward s equals. R. Tittel we just apply the formula, but, uh, careful steak assistance with the units. You need to express this in radiance because remember, an angle has been defined by racial distances. This is to be explicit radio, So first you convert it into Iranians. Now you know that radiance is going to be smaller than Dewey's says when you pi over 1 18 But the blood i 100. So you could even simplify this and say It's just pie divided about 1.8. That's plug it in s equals five multiplied by pi, divided by 1.8 calculator 8.7 three Nieto eight point selling readers. Let's check in. This answer makes sense. This is five meters. This would be 100 degrees. Has it been more than 90? So how many times you find this in this arc? So I find it once and I've been more than 1/2 So 1.61 point seven thanks
3. Trigonometry Basics.: What I'm going to require from you now is a little bit of imagination. But it's worth it because it will make trick really easy for you. Imagine the ground now imagine a war perpendicular to the ground. Now imagine a plank. I wouldn't plank could look like this. The length of the plant would be our The thickness of the plant would be negligible, so close to zero and the depth of the plight. We don't really care because we are going to look at the plank from the side. Now, imagine placing this plank and the corner here, attached to a mechanical system that allows you to move the plank will change its orientation like this. The finding an angle did that like right now. Imagine light. A beam of light come from the sky on all the beams. Airpower. You see what happens. The plant is blocking some of the beams of light and this result in the shadow. This shadow is a projection off the plank on the ground. It's like a little bit when you do these things little fingers, you know, and you have to make some shapes using the shadow you're projecting the shape of your hand on the screen Here, the shape of the plank is projected on the ground and results in the shadow. What would be the length of the shadow? It will depend on the sides of the plank. Yeah, If the plank is large, where the shadow is love, it will also depend on its orientation. Is he? If the angle here is not well, the shutter will be not so long. But if the angle is small, the size of the shadow will be larger. If the angle is zero, the size of the shadow will actually be the length of the plank. The lengths of the shadow depends on two things. Length of the shadow depends on the length of the plant and the angle. The orientation of the plan. The length of the shadow is equal to the length of the plank, which applied by the cost sign of the angle. Now I have a little mental techniques Way to remember when I want to projection off the plank on the ground. I need to squash Yongle and this is what I'm doing. And Scott sings costing the angle. You see what I mean? I'm playing on words. It's a bit like today practical because I suppose you guessed now that the plaque is a factor. So if you represent the Axis X and Y and the victor, Uh, yes, and there's an able to tether Well, when I want to know the projection of our X, giving me the X coordinate off our I squashed the angle. In order to get this coordinates, I get X of our equals costs off Twitter. Imagine now that we have a beam of light that goes always entirely, so it hits the wall. When it reaches, a blanket is blocked by the plank leading to a shadow, this time on the war. That's kulig lengths of the shadows on the wall. Well, the length was a shed on the world will also depend on the length of the plank and the angle. So the lengths off the shadow on the wall is equal to our multiplied by the cost sign off the angle that I'm squashing. So here is a complementary angles. It's 90 minus 10 but 19 minus data. Is scientist applying to our vector. Projecting off on the Y axis will therefore be also ended up and this is why coordinate? Why this really practical? Because you do not need to use a triangle system when you see a vector and you want to know its projection or its coordinate on the maxus. If the angle that you have is in between the victor and the access is quashed, it costs angle. If it's not meaning that it's the complementary angle when you sign it. This representation with the plank on the shadow being the projection of the plank on an axis like elector projecting an axis giving you the coordinate off the specter on this access This is practical because you do not need to use triangles. Yes, If I want to find the X coordinate off this vector I squash younger, I cost you. If, on the other hand, I want to coordinate on this access, Yeah, we were having a complementary one, so I signed it a little. I just need to remember on go between access and vector costs on go note between access and Dr Sign, we're still going to have a look at this triangle thing because actually, we're going to reconstruct the formulas backwards. You'll see I represented here are vector are a wooden plank. All you wanted was an angle to Tavis his ex. And we can also have the X and Y coordinates eggs off r equals R Kosta and why off r equals r sine theta He recognized triangle rectangle triangle. But they do represent here the hypotheses as he quoted the magnitude of the vector. If I put the angle here, Victor, the adjacent to the angle is equal to the X coordinate. So our cost ETA on the opposite is equal to the y Coordinate. So our scientist, just in case you didn't see it, this side is equal to that. This side is the y Coordinate the opportunities of the rectangle Trying always are our appears here. So I'm going to replace the are hereby I puffiness giving me opposite. Of course I put the news but applied by scientists and that can be arranged. This scientists equals opposite of my partners. You know this formula I know you know this formula. Then you can do the same thing with the adolescent a distant equals I put the news we planned costed that because it would place the are by I put this beating me cost it R equals I doesn't on hypotheses. And finally, if I divide one equation by the other like this, you can see the I put the news goes away and I get opposite on adjacent equal Santa on Costa leading opposite on adjacent equals 10. So yes, third formula, which is Tanta equals. That's how the opposite on I just He already knew these formulas, I'm sure. And now you know where they come from from the Cartesian coordinates living. And if you want to play a little bit, let's inject a little bit of Peter going there. Yes, we have that rectangle trying. So you know, they the apathy squared equals to the sum of the squares off the side. Let's do that. So I will write. You have space? Yes, I put in your squared equals opposite squared. Plus, I just don't squared. Opposite Squared is R squared sine squared titter How just in squared is all squared Cost its web. You see, I can fact allies the r squared cost quite theta plus sine squared did the But you know what cost quite a topless sine squared two days it's one. So this gives you r squared the iPod thingies square down square. So the iPod thin it is our playing back and forth like this between the triangle on the coordinates of a vector kind of fun. Most importantly, it allows you to understand what's going on.
4. Training exercises.: the sun is low on the horizon. The light ways make an angle of 20 degrees with the ground A man one meter 80 totally standing straight. What will be the length of his shadow? I would present it on bold man like a vertical block off one meter, 80 height And also the raise the light lace going to with the man and they blocked by the man. But this one? No, I've just above the top of the head of the man and manages to pass all those above will pass and hit ground. So from this point, you have light. That means that here we have shadow. We also know that the angle made by the way of light with the ground is 20 degrees. What are we looking for? Well, we're looking for this length the length of the shadow. I suppose you recognize a triangle here, Elect Angle triangle, where you have one meter 80 here. This is what you're looking for and you have the angle 20 degrees. So what form should I use? Tangent? Because tangent titter Izzy quarter the opposites on the adjacent. So if I put numbers in tangent of 20 is equal to the opposite 1 80 divided by the length of the shadow. So the length of the shadow is 1 80 Did I? Did I turn off 20. I find 4.95 meters. That's about five meters. Maybe you noticed shadows becoming longer and longer when the evening approaches. Yes, the sun is getting down on the horizon, so the ways of light I'm making an angle with ground, which is smaller and smaller. Imagine here. If there is a way of light making a small angle, the shadow will be the whole spools. A 30 kilograms sled. The tension in the rope is 200 new terms and makes an angle of 15 degrees with the ground. The fiction is 150 new terms. Calculate the horizontal acceleration of the sled. I represented the 30 kilogram sled lack of blocks. I also look presented the forces which country beauty, horizontal motion, attention and fiction. I want to know what acceleration has the block in the horizontal direction. So I need to find the net force in the always enter direction that is a always into a component of the Net force F Net and the extension. First I need to define the next axis. That's how countries positive in the demolition emotion there for this one. Then I will project the different forces attacked on the block on this access. What is the contribution of tension on the X axis? Well, it's it's projection. It's actually it's component on the X axis. This here we know the angle it was given in the text. 15 degrees. Attention contribute to the Net force by t costs off 15 because I'm squashing the angle to get the projection of tea on the X axis. Let's look at fiction. Fiction contributes fully, but in the negative direction sets minus F that gives us how much was tension. I do not remember clearly, Newtons. I cost 15 minus and fiction was 1 50 I find 149. I know that the net falls in. The extradition is equal to the mass but to plan by the acceleration in the X direction, which is what I'm looking for. So I mean range F net X over M A X equals 1 40 divided by 30 which is 4.7 meters per second squared. I'd like to make a comment on this. When we rode down the X component of the Net force by adding the contribution to the X axis of each of the forces, we actually projected each of these forces on the X axis, for example. T we projected t on the access giving us t cost 15 with question angle. But when we came to the fiction, we actually just said, Oh, the friction is fully contributing because it's in the same orientation but in the negative direction so minus f without thinking more what we're actually doing. We actually also projecting the fiction onto the exact sis with an angle of 1 80 degrees like tension. We squashed the angle with fiction. We also squash the angle, but on a group of 180. Actually, what we were doing, we were doing this. F X equals tickles. 15 plus F costs 1 80 but course when 80 is minus one, giving us t cost 15 minus F. So when you have a force of fiction going against motion or any force in the negative direction, you tend to put madness and the magnitude of the force. But we'll do you actually doing is putting the magnitude of the force and projecting it, and the positive dying should through an angle of 180 degrees. So post by cost. 1 80 giving u minus force that was not a condom. A plane takes off from City A and has a nascent with a 30 degree angle for two kilometers. Then the plane remains of his dental and travels for more kilometers. The decent phase occurs with a 45 degree angle until landing. What is the height reached by the plane after the ascent? I'm second question. What is the distance between city A and city? The first question off this exercise was to find out what height the plane travels resentfully. So we're looking for this height. That's great X. I supposed you recognize the rectangle triangle here by the night Kathy News off two kilometers, an angle off 30 degrees on the opposite side, being what we're looking for, which for me, actually, I use where the one was opposite, and I put the news a sign off. 30. He whose opposite over i prettiness so X equals two. Signed 30 Sign 30 is 1/2 so X is one kilometer. This is one kilometer. The second question is about finding the distance between the departure, the arrival, this distance. I think we D is This science is composed off this one, this one and this one. Call it D one D 21 defeat So D equals D one plus D to and the three that we have to do tonight. Let's start with the one I could actually squashed the angle by bringing down the APA thinners that would give me the one equals two costs or 30 cost of 30 square with a fever to sits squarely to three kilometers. Detour is quite straightforward. It's four kilometers. What about the three? But we don't have the angle, but we also have the height here. X 13 meter. We had it here also in this re Congo Triangle. We have the opposite side on the angle so we can use tongue to find the adjacent So 10 off 45 isn't to be equal to opposite of adjustment. So one of the three, but I know that 10 or 45 is one. So the three is just one kilometre. So give me that now if I won't be, I just adul these and get five class quit of three kilometer that's about 6.7.
5. The Unit Circle: the unit circle. You've heard about this guy, right? Well, he's about to become one of your good friends. It's a great tool, very practical in many situations. The last to understand visualize trigonometry. So first, I'm going to try to draw. So how does it look like? Not too mad. And now I'm growing the access. Thanks. And why now is to choose a point em on the circle. So remember what we did before If I chose a point and on circle centred around the knowledge Oh, so I would have a vector om here off magnitude are. And in that case, the components off this vector would be our Kosta. When ties the angle there and here are signed it up I'm going to give a value of one to the magnitude off Victor Om. So now the radius of my circle would just be one. And here are is one. So the components of the vector around will be just costed are and scientific. That's what we call you a unit circle because God radius of one. You can also think about tangent. Tensions off Tetteh is in quarters scientist up. Of course, if you know that right? Well, here it corresponds to why next. So tell Tetteh is equal to the why over the vector aware over the X off the victor of them . Well, that's Ah, that's a slope. This gradient Tanta here is a gradient off the line of them. You noticed I drew it again much bigger this time because I'm going to put lots of numbers in it. What we're going to do, we're going to try to evaluate the values off course and sign off different angles and you will see a pattern, a pattern that allows you to remember them. This table with cost 30 cost 60 costs 45 you have to remember the numbers well, no need anymore this school and then go 30 degrees. So you see that this value will be cost 30. It's actually a square root of free of a to which is around 0.9 and you can see it's about 90% off the length here in the majors, but still now and I go 45 degrees. So if I projected him about 2.7 70% of the radius value is actually square root of two of two, which is around 27. Now let's drew on and go off 60 degrees cause I know 60 World War 60. They don't have much space left. 60 cause I know 60 is 1/2 so square root of 1/2. That's one home. 15 They have costs off. Zero. Well, that just gives me one. Take a black. Depend for this and cost of 19. That gives me zero. We have this list of numbers zero square root of one of the two squared to to to It's going to fever, too. And one and the values were Values are 0 1/2 and that's about do seven. A soup or nine, and this is one. If you have an angle to say cost 60 in an equation, just visualizing in its circle, drawing uncle 60 degrees and see Wait ends up. If it ends up around 0.5, then you know it's 1/2. If the angle you calculating ends up at 0.7, you know, school to to to If it ends up as you 0.9, you know it's quit of 32 This is the only table you need to remember. And Comey Yeah, let's sign. Well, look at the values of sign for 60 degrees. It's great to see you, too, Mrs 0.9 45 degrees. It's close to two, and for 30 degrees it ends up in 1/2. So scared of one of two. So something. You just visualize the angle in your head within the unit circle and you see way lands on the X axis. Because on the Y axis design now is the angle is bigger than 90 degrees. It's the same principle. I only have these colors to black. Suppose I have 210 210 is 1 80 plus 30 so you can draw an angle of 30 degrees. Here, that's going to be 210. Where does it arrive on the X axis? 0.9 So and it's negative side course of to 10 with me minus square root of free of To what about? Sign off to 10? Well, I just projected on the Y axis. Arrive around 1/2 on the negative side. Say it's minus 1/2. Let's do another one say 300 or 300 just 300 so What is 300? Well, 300. We can see it as 360 minus 60. So it would be minus 60 all you can see the turn and 70 plus certain it's 91 80 to 70 plus 30. So that would be your 300 degree table like this. So what would be the cost here? Where is here? The cost new project on the X axis you end up at one house would be the same. Well, if you project on the Y axis, you end up at minus opponents of minus square with 32 and that's how you feel it out. Train on this next time you're waking on an exercise which involved calculating some co signs of subsiding and the angles are pretty standard. 0 30 45 60 or 19. Don't you see you guys later trying to figure it out by yourself? Use a piece of paper, draw the circle, draw the angle on, estimate the value of the co sign or the sign, and you know it can be only one of these values. It is also possible to estimate approximately the value. Often co sign was signed off another angle, then 34th Final 60. Suppose, for example, you have 142 and you need to find co sign off 142. What is the closest angle for which you can't buy the coastline in san with this technique of 1 42? Well, 1 42 years between 1 2150 is quite close to 1 50 So you draw 1 50 and if you do 1 50 you find a co sign of 1 50 minus quite a bit of three or two. My squint of Syria to is no exactly my simple 990.866 something So you know that 142 degrees we'll end up here before, little less for something probably like minors. You point a let's check it out. Good sign of 1 42 Not bad minds upon 79. Chain on it. It's very useful. Believe me. Once you get a good off, this trigonometry becomes much more simpler as an exercise. Try determining these values without using your calculator. Signed 30. I draw a 30 degree angle on a project on the way Access I'm goes to 1/2 So it's 1/2 sign. 30 equals one, uh cost 1 20 So 100 twenties 90 plus 30. So that's my 120 angle costs. So project on the X axis and I get about minus one. How so? Cost 1 20 is minus one home cost to 40. So to 40 is 1 80 plus 60 said lives here. And when a project on the X axis Yes, it's cause So the X axis I get minus one, huh? Cost to 40 is equal to minus one 10. 45 federal 45 degrees, as a naval tell is, Sign over call sign. So the term in san In Kasai Hey, I read soup 17 for the sign as you 0.7 for the course sign Say it's closer to two and the same value for both. So it gets quote of to to define the square root of two of two. Tell of 45 is one 10 off 60. I drew an angle of 60 degrees on same thing. I find the son of the coastline. So you see, the sign is going to be something that's growth of three of to While the co sign, we'll be 1/2. So the time is silo co sign so half a square we to see. Divided by 1/2 it's close to three. 10 of 60. It's over 23 Divide by two, divided by 1/2. The halfs goes away. It's transcribed. Sign off! 330. Fear of 30th here in 60 miles. 30 self. It's my 30 degrees and I'm looking for the sign. So it's on the Y axis. I need to project my vector and I find minus one off. So sign off. 330 is minus. One. Sign off 57. Where 57 is around 60 so I can draw something like 60. I find score of 32 so it's going to be led to less. That's going to feel, too, because it's not exactly 67 Toby less so you see, I was here course off. 480 480 is 60. Class 1 20 so it's actually costs off. 1 20 which is minus one house. Sign off Pi over three we're using. They didn't this time. Remember, half circle is pie, so pi over three. It's going to be the half circle, so 180 degrees did I by 3 60 degrees. So we're looking for the signs. Se is going to be square. Bit of three or two tangent off repair before he paid before is 3/4 off. Pie, right? So it's 3/4 over half circle. It's Tanja says in the sign of a co sign for the sign we find Scoppetta Tutu. It's about 70% off the wages and for the coastline we find minus screwed up to over two. So the same number for both but for the coastline is negative, so tangent will be equal to minus one. So it's close to two. Divided by my a square root of 22 two in minus one. Yeah, the stroke is negative cause I know 17 pi over three is 18 pi over three, minus pilot three. What is 18 private three 18 very by 36 expired. So it's free circle. So seven Power three is actually just minus pi over three. Let me just show you 17 pi over three is 18 pi over three minus pi over three but £18.3 is just six pie. That is three circles. Two pi, 4.6 by and then minus pi over three. So we're looking for the co sign. If I project it on the X axis, I find something like, Well, half no sign of turning 65 degrees. 0 to 65 is not far from to 70 Purchase. Now he 18 to 70. So it's actually yeah, so co sign of 265. You know it's going to be a negative value on baseball. Let's check in on the calculator course. I'll up to 65 my last few points you
6. The Unit Circle (advanced): the unit circle is a bag full of magic tricks. The unit circle is a bag full of magic tricks. He's got more up its sleeve. You know that, right? I've got em. Which is a point on the so called radius One. And the coordinates off this point n will be Augusta for the X coordinate. And scientists are for the white coin. Good course and signed a treaty Limited functions. When you change Stater while the value change well, you can put as a trigonometry functions in there. Did you know that you could put tangent in it? Did you know that you put second. You can put Kosek cooked engine. You can put all of them in it. Yes. You just need to know one thing. The theory Mawf palace. So I'm going to try to explain its space here. Yes. Imagine trying. So the triangle has I put the news on two sides. Now I cut to triangle with the line like this parent, all to one of the sites. So it ends up with having kind of to try and going to one. I got this triangle. I've got a smaller train. Well, the Palestinian Well, tell me that ratios offsides off each off. These triangles are equal. Meaning that the ratio these two guys will be equal to the ratio of these two guys here, for example, 5 to 1 in there. Three. That is 1.5. Because to over one is 23 of a 1.5 is too. But it works also for the other sites. Any sides? Actually, the ratio off this and this is equal to delay. Show off. That and that. Oh, the ratio off. What else you have? Yes. You know the color Les, show off this the party news over the adjustment is equal titillation of the empathy news off the address and bigger want. That is to tell a story. And we are going to you step. I'm going to extend this this line and I'm going to take attention to you. Detention Teoh Circle. And maybe now you recognize the same configuration. These two triangles. I want to know this length. Hey, going a for now. So a over this length in this triangle is going to be equal to that over that in the small trail which is scientists of Acosta. Well, that's stunted. Egg was tended in red. Here, this length is just 10. Did that. We can continue to play with this triangle thing. I want to know this leg That's going to be so. I can say, for example, be over one. So this side and the big trying divided by this one is equal to this side divided by this one. But this one, this side is one all right. Moving from CIA. This side is one. It's outrageous and circle divided by cost data. So be is one of cost data set the pretty cool. So this we have in blue here. This beeline is sick. Can you tell me what is a green distance here? Get to shop. Look at the triangles. We got this big triangle and got this small trying with it. Give it a shot. We have to use a Castillo. And again we quit. See? All right. So we would have seen over one see of one in the Big Triangle is equal to this over that. Well, that's cost ITER over Scientific Acosta of a scientist which is so see it is a good time posted. What about this length? But It's starting to become a mess on my bold this link. What would it be? Let's call it well here the triangles. This one for the big one. This one was a small one. So we can write down of space somewhere. Yeah, yeah, maybe. No. Maybe they just here we have d over Could can deter so d of coltan titter will be equal to this length divided by this one. So that's one because it's a radius of a circle divided by Kosta. So that gives me d equals Kutan Teta on Costa. But could Santita is cost iter divide by scientists with blood by one of Costa The casita councils away yet one on scientists, which is Kosek done So the is constructed. This is constructed cook sink did that pretty cool? Try to do it again. Tried to find out all these relationships It's a good way to visualize trigonometry geometrically in your head. This is the last episode off this course We have worked together on straightening up your algebra. The mystifying vectors, which is so important in physics and reviewing the trigonometry notions you need in order to approach physics exercises and example, is confidence if you follow this course seriously and went on all the exercises. Congratulations. You just made your life at the physics student so much easier. The only thing you need to do now is to enjoy this fascinating topic. That is physics. Yes. From the formation of our universe to the mysteries of the infinitely small. Lots of wonders lie ahead of you. And what is cooler is that this voyage will not be disturbed and more by knowledge gaps. Not so. Enjoy the trip. See you out there.