exact value of sin(3 degrees)

today work on the fund it's a for you for sine  of three degrees and what do everything from   scratch including all the special red triangles  and also the angle difference formula that we   need so be sure to check out the whole thing this  is going to be really cool but before I start let   me first give a shout out to chest her because  he recently did a video on sine of 10 degrees by   using complex numbers it's really cool this really  gets got checked out a video if you haven't done   so already and I also want to thank Toby for your  t-shirt thank you for sending me this t-shirt this   is ketchup this is a kind bottle and her channel  name is Phoebe's if you guys don't know it already   I will have the links to their channels in the  description be sure you guys go check them out now   let's talk about sine of three degrees everybody  knows one plus one plus one equals three by no 18   minus 15 is equal to three so let's look at sang  of three degrees as 18 degrees minus 15 degrees   and the reason is because both of the 15 degrees  and also the 18 degrees they're special special   angles you'll see why anyway as I promised that  I will show you guys the proof so the first thing   is that I'll show you guys how to prove a angle  difference formula for sine and this is how we'll   do it we'll use complex number we used all this  formula so here is the deal we are going to look   at the following so consider e to the I times  alpha minus beta power like this first in first   whenever we have e to the I theta and ingo we can  use the Euler's formula and Toby has a really nice   video on the Euler's formula so be sure you guys  go check that out here just use the formula well   what this is is that we will just get cosine  of the angle which is alpha minus beta and then   we add I times sine of the angle which is alpha  minus beta like this okay so this is nice now we   can look at this in the following way because of  course we could distribute the I and then because   this is exponent so we can do the following we can  look at this as e to the I times alpha and then   times e raised to the I times negative beta power  like this when you multiply this of course you   add the exponents of course you can factor out the  eye and you get that and now you will see when you   look at this we can apply the orders formula for  the first part and then the formula for the second   part this is very nice so if you look at this part  we get the following this is going to give us well   I'm going to just continue right from here because  otherwise I would have space to do all the work so   I will probably you write yes 1 & 2 so number  1 is right here and of course 1 is equal to 2   I mean expression 1 is equal to expression - ok  anyway here's the deal for this one or the spume   aligning action we have cosine of the angle which  is alpha and because I only have planning going   side I will just write it as cosine alpha without  the parenthesis if you see this way and then we   add I times sine of alpha that's nice so that's  that now for this one we'll multiply we have the   angle P negative beta so here we have cosine of  negative beta and then we add I times sine of   negative beta like this and this is also very nice  now we can simplify this a little bit here this is   equal to cosine alpha time a me plus I sine alpha  and then this right here because cosine is an even   function so cosine negative 8 is the same as  cosine beta so this is cosine beta and now I   don't need to put on a parenthesis because I don't  have one thing but signs an odd function it's okay   though that means we can just put a negative in  front so we have minus and then we have the I   times sine beta and because now just update I'm  not going to put down parentheses otherwise I'll   go crazy so this is what we have and now if you  were like of course you can continue to multiply   to the star and that's actually the strategy well  keep in mind what we are trying to do is we're   trying to get a formula for sine of an angle  minus the adder which is right here isn't it   well look at this expression we have a real part  and also the imaginary part and we know one it's   equal to two so if we can get the imaginary part  fun expression 2 that means that has to be the   same as that that's so nice so here is the deal  I'm just going to multiply out everything real   quick so this times this is cosine alpha times  cosine beta and then this times that is - that's   put the eye first we have eye and that's right  almost line first so we have sine beta and then   cosine alpha and then we will have this times that  I will put it down here which is plus I sine alpha   then cosine beta and then this times that negative  times a positive is negative and the items I is I   square and then this times that is just sine alpha  times sine beta so this is what we have well from   here as you can see this right here is a real  part because that's a half the eye this right   here it's also the real part because I squared is  the same as negative one so this is just as real   as we are but this and that they both have the I  well the imaginary part on the left hand side has   to be the same as the imaginary part on this side  right so they are technically equal 1 is equal to   2 as I said oh by the way photo - Donald - are  here there we go all right so now we know that   this has to be the same as that we ignore the eye  right because we're just looking for the part so   therefore I'll just put down so we know that sang  of alpha minus beta let me just put this down here   for you guys and yes you guys can do the same for  cosine but I just care about the sign right here   sine of alpha minus beta is equal to this and  that but that's put on a pasty first which is   this part this is equal to sine alpha times cosine  beta and then we subtract sine beta times cosine   alpha cosine beta and then cosine alpha like that  angle difference formula for sine again you can do   the same thing with cosine but you had to pick  up the real part found this right here okay so   now that's pretty nice well we continue this  right here is nicely equal to sine of 18° so   let's just put that down because that's the first  angle 18 degrees and then we multiply by cosine   of 15 degrees so cosine of 15 degrees right here  at the minus sine of the second angle which is 15   degrees and then we have cosine of the first angle  which is 18 degrees so that's just ready down here   yes when the world is dying 18 degrees and also  cosine 10 degrees right so now let's talk about   a special special triangle this is done so that's  nice okay so let's put this down right here let's   see if we can fit everything right here I don't  know but we'll try this is how you generate this   is how you construct this is how you come up with  stuff 1872 90 special right triangle but I don't   want anger to be 18 years so this is what we do  so you first draw I thought this triangle and   because we are constructing the right trying the  triangle you can just say the sites are 1 and 1   right here right because we are just constructing  the triangle from scratch and you want this angle   to be 36 degrees again no 18 yet if this is 36  and this is a salsa this that we know this to   INGOs will be the same for the pace try to play  singles and of course 180 - 36 and / - you know   that this two angles will be 72 degrees and also  72 degrees that's very nice well you guys should   know / - next if we can figure out this side cut  in half then we can get a special right triangle   isn't it but I don't know what this is yet but the  best strategy when we encounter something that we   don't know it's just that we can call this to be  X how's that very nice huh now check this out well   this right here is X what will do is I will just  kind of rotate 36 degrees so I will actually cut   this in half because this is 72 degrees and if  you rotate 360 degrees this is exactly half of   that angle angle by setting right well this right  here is 36 degrees and notice this is 36 this is   72 that's also 72 and with that said this one  gets X this right here is also X now here's   the deal if you look at this right here we have  this triangle in black Dementors try to make it   as pale as possible we have 1 1 and X but if  you look at this red triangle okay we have X   and X so let me just put this down this is older  and notice this is 36 this is also 36 right so   they asked the model because all the ink costs  are the same but this is X this is X I need to   know for this is if I can do so I think can come  up with a really nice similar triangle and most   some equation for that right well check this out  the whole thing is what and this is X already hmm oh yeah I cut it this right here it's X right  well check this out this right here is 36 degrees   because we said it this is 36 that was 36 that's  how it can make the 72 and now this right here is   X this is 36 that's also salty 6 that means we  have isosceles triangle so if you look at this   part since this rep is X and this is X also this  that means this right here it's also X as well   with that's that because the whole thing was 1  this is X so this red part is 1 minus X so I have   to think there and then of course you can just put  this down right here 1 minus X for the base now as   we talk about it these two triangles are similar  so we can do the following so I will just do this   I will say 1 all for X its equal to I'm doing this  over that so I will put this over that it's equal   to x over 1 minus X like that and now we have an  equation right here let's just go ahead and stop   this equation real quick because multiply if you  were like we get x squared that's equal to this   time stamp which is 1 minus X and then you put  everything on one side this is x squared plus X   minus 1 and now we'll keep a 0 and then of course  we can solve this by the quadratic formula so we   get X is equal to negative B which is negative  1 plus no minus because if you have a minus the   whole thing would be negative you wouldn't make  sense in a triangle situation so just the plus   versions all we care and open the square root 1  square right that's a P Square so we have 1 minus   4 times this and that but this is negative  so become plus so plus 4 so 1 plus 4 there   are here in size 5 and divided by 2 times 1 so we  have divided by 2 so this is what we have and yes   this is some what they did to the golden ratio  isn't it but anyway check this out what we are   saying is that X is equal to this number so they  may just come back right here for you guys X is   equal to depends on how you want to write it let  me just read yes negative 1 plus square root of   5 over 2 like that well well if you were like of  course you can look at this and what we do is of   course you just cut this into half like that so on  all we can produce we can come up with the special   right triangle that we need so of course this is  a right angle originally was 36 we cut in half so   this right here is 18 right so I will just tell  you guess we have this right here all right so   now Joe Giusti okay so this is 18 degrees okay the  side is 1 well we can have so I'll just put it the   other two in the denominator 2 times 2 is 4 in the  denominator so here we have negative 1 plus square   root of 5 over half of that so we have to divide  it by 4 like this based on this we can figure out   sine of 18° because science the opposite over  hypotenuse so it's just this number pretty much   so we get this right here but we don't know what  cosine is we need the adjacent side so let's see   if we can come up with the adjacent side right  here and to do so we'll just use the Pythagorean   theorem but you're not famous ones I'm not going  to prove that a squared plus B squared equals C   squared but I will label this as I say B so to get  this type what we do is just open the square roots   happen your square which is one square right one  squared and then minus this thing squared right   so let's put it down like this one squared minus  this x squared which is negative 1 plus square   root of 5 over 4 and we are going to square that  this is going to keep us that side and now let's   just do the algebra here real quick this is square  root this is 1 - okay the bottom here is of course   16 4 squared 16 and on the top that's C make a  square with slightly bigger this is a binomial   squared so you have the first thing squared which  is 1 and then you do 2 times this and a so it's   going to be minus 2 times this and now it's just  2 times the square root of 5 and then last you   add this thing squared square root 5 squares just  a nice 5 and now of course we can get some common   denominator we can just put a 16 here and 16 here  and we see this is the square root everybody over   16 like this right on the top we have 16 minus be  sure we have a parenthesis now this is 1 times 16   which is 16 16 minus 1 is 15 15 minus 5 is 10 so  we have a 10 right here and then of course minus   times a minus is a plus so we have a plus 2 square  root of 5 like this and if you would like and just   simplify this real quick by 2 divide this by 2  we get 8 divided by 2 we can 5 divided by 2 this   is just 1 so this side is square root of 5 and  yes we have a square root inside the square root   it comes up really often when we are doing sine  cosine by anyway we have this right here and then   divided by 8 and of course you can simplify that  but that's not where your part that's later okay   so ladies and gentlemen here we have a special  special right triangle for the 7t to right this   is the 72 degree 18 degrees 70 to 90 special right  triangle right yeah whoo okay so far so good huh   now we'll come back here okay I don't know what  cosine sine 15 saw okay don't worry I have another   special triangle for you guys so here is the deal  but that's just that's just actually your finish   everything in the middle huh in order for me to  come up with the 15 special special triangle we   have to do the regular special triangle first so  anyway this is how we do the special train goes to   normal one okay so when you start with a square  this is one one one one one one everybody is 45   degrees I mean everybody is 90 degrees and you  just cut this into half and by Pythagorean theorem   this is the square root of 1 square plus 1 square  so this is very nice so if you just make the cuts   we get this right here yes 45 degrees 45 degrees  special right triangle isn't it this is one this   is one the diagonal is the square root of 2 which  is the hypotenuse so this is the crucial square   this is the usual special right triangle and if  you start with a equilateral triangle everybody   has the same sir everybody has the same angle so  just put on one and everybody is 60 degrees inside   and this is much better than that one because all  we have to do is just cut it right now middle in   that case this is one half and we can again use  the Pythagorean theorem we'll see this right here   is one right here and then this is one half here  and if you want to figure this out in fact the   easiest person is you don't start with one you  start with two so I'm sorry let me just erase   this a little bit start everything with - so what  constructing the triangle so I can put down let's   say we want each side to be two and then we can  say this right here will be just one so we see   that yes just two and this right here's one but we  still have to figure out this side which you can   do it the quick way this right here let me just  indicate that right here which is a square root of   the hypotenuse which is two square right and then  minus this size square upon nu square which is 2   square minus 1 squared and this is right here is  4 minus 1 which is 3 inside here right so we have   square root of 3 like that and this is the third  he because that's half of 60 30 degrees and this   is still 60 and we have a right angle so that's  a special right triangle well similar but this   is much cooler but of course somewhat difficult  as well this is much easier now 15 75 90 special   right triangle this is how you do it you are going  to utilize this and that let's start with what I'm   going to start with a 45 45 special triangle okay  but I'm not going to label the sides yet why we'll   do it I was just first make a cut right here or to  a line and then our 12 45 degree and they can work   constructing so this is okay again this right here  is 45 degrees I'm not going to label any sites yet   then why will do is I will put this right here  I just want to stir it decrease to be right here   though so when we do that I will have 30 degrees  here okay and then I'm just going to continue my   lines like this but why would like to do this  I would like to start right here and then make   a right angle so this is a straight line I can  just put crying go here okay and that's pretty   much it and then I will finish this great angle to  do so of course I will just you know it's dan from   here to here and then I will go up right here and  I will complete this great angle not complete the   square this is so beautiful because you'll see  this is 45 this is 30 and of course everybody   right here at ought to be 90 because we have a  rectangle and of course with the set this right   here will be 15 degrees you can do this in the  head 90 - 45 - 30 you get 15 degrees so now we   just have to figure out the other side and notice  why the earlier because we're constructing the   triangles on our own we can label wiper set that  we want but once you put on some numbers the other   numbers have to based on the numbers that you're  chose right this is how I want to do it look at   this triangle this is 30 90 and this right here  60 isn't it I would like to put out the special   right triangle right here so I would label this as  1 and the hypotenuse which is this right here our   label does - this is this die and this right here  square root of 3 so put this down square root of 3   right here okay so now here is the deal this is  45 45 90 special right triangle but we know the   hypotenuse in this case is square root of 3 well  earlier the HEPA news is show biz square root of 2   so how can we fix that don't worry because all we  have to do is multiply by square root of 3 divided   by square root of 2 so let's just multiply every  side by square root of 3 divided by square root of   2 when we do that you will see these are now pre  cancelling out and we have just graduate of sweet   as the happen news and we can of course kind of  scale this up a little bit this times that is just   square root of 3 over square root of 2 likewise  for this side square root of 3 over square root   of 2 and this is very nice and that's exactly how  we're - right here I will label these pictures   here this is square root of 3 over square root of  2 likewise this is also this so that's very nice   but the do is that you see this is 90 this is 45  and the 4 straight line so the good news is that   this yeah it's also 45 degrees so is this because  again we have a right triangle so this right here   it's also 45 degrees so this is so much easier so  much so it is so nice because in this little right   triangle we had the hypotenuse is equal to one  well this type I will do is I will just divide   everybody by square root of two so if that's that  this part is 1 over square root of 2 and this part   is also 1 over square root of 2 that's so nice  that's still nice so nice now here is the deal   check this out yes we have the 15 degrees here  huh check this out 15 degrees and this is 19 well   I need to figure out this die I can just look  at here notice I just had to do this plus that   thankfully I didn't rationalize the denominator  because they have the same thing with it well if   you Russian I still have the same denominator  anyway one choose weight sorry this is square   root 3 over square root of 2 anyway I'll just do  this pasta for this side which is square root of   3 plus 1 over square root of 2 and then for this  die as we can see the whole thing supposed to be   square root of 3 over square root of 2 we have  1 over square root of 2 right here already so   this right here has to be this minus that which is  square root of 3 minus 1 over square root of 2 and   congratulations we just get our special special  right triangle for the 1575 90 special retract   special special right triangle ok so we can come  back here and do all the things that we need sine   of 18° let's look at this picture which is this  all for that so it's just this because thankfully   this is equal to 1 therefore I will just open  parenthesis some of 18° just this which I am way   all put down the square root of five first because  this way has a square root of streak first I think   we just do that so square root of I minus one  be sure the square root is just for the 5 over   4 right and over 1 doesn't really matter so that's  sine of 18° and then cosine of 15 degrees is look   at this picture we will have to do adjacent over  hypotenuse and don't forget hypnosis 2 so we had   to put a 2 in the denominator so we have the  numerator being square root of 3 plus 1 over   2 and then there's another school to write so 2  square root of 2 like this so this is very good   and then minus and saying of 15 degrees same  deal we do this over that which is square root   of 3 minus 1 over 2 square root of 2 and lastly  cosine of 18 degrees cosine 15 degrees is adjacent   on here divided by the hypotenuse which is just 1  which is just this number so I will put this down   also in blue why not this is sandy multi-source so  I was simplified this in your head okay so that's   the students together what up we have square root  of 5 plus square root of 5 so that's ok yeah /   well let's do kiss this right here this is square  root of 5 plus square root of 5 Banna divided by   square root of 8 for that right and square root  of a we know this is the same as square root of   4 times square root of 2 so this is just a 2 and  this is another square root of 2 like this oh my   god in the end let's see if we can simplify this  a little bit or not not so much though let's see this is 2 this is 8 this is a this is  square root of 2 hmm what can we do   nothing too much huh so perhaps  this is 4 times 2 that's 8 then   that's 8 and the square root of 2 now  I should care our 16th and we're in   the denominator let's see square root of 18  is no columns 18 is Gerudo yes yes and then oh I see let's just brush on I study now that's  not seem too much we can do so that's just do   that so I want everybody to have these two things  to have 16 the denominator okay because this is a   square root of 2 so let me just multiply this by  the square root of 2 and that's also multiply this   by square root of 2 with that said you will see  okay let's put this down right here and because   this is the final answer I was just ready big  enough for us to see so 4 times that's 8 times 2   is 16 so I can have that and now I'm not going to  multiply all the top so you can do do that only if   you like this is square root of 5 minus 1 you can  do you can have the square root of 2 somewhere but   you can also distribute this just your preference  that's just at least multiply this in how's that   yeah so we have square root of 6 and I think  this is how Chester has it your PDF on this PDF   as well so I saw Chester this is the phone that  you have all right so distributive square root   of 2 so that's nice now - in between but check  this out though square root of 2 times guru 2 is   2 times these times dice Oh to get is just 1/8 I  need to match with 16 so I need to multiply this   by 2 this by 2 so I can get a common denominator  6 teen so be sure we have this tooth up to down   low to auto in the front and then I will just keep  this that's how they are in I guess this is square   root of 3 and of course you can distribute that  we feel iPad that doesn't matter anyway what I   know so minus 1 and this is a bizarre square root  square root of 5 minus square root of 5 so ladies   and gentlemen sine of 3 degrees is this right  square root of 5 minus 1 times square root of   6 plus square root of 2 and then minus 2 x squared  - 3 - y pense five plus square root of 5 all over   16 right Wow I fitting everything on this poor  I'm so happy about it no this is so satisfying   but anyway this is it think this has so much for  watching I should prep more to organize the port   before I do this but this is so cool anyway if  you guys liked the video please give me a like   and again please check out Chester and also Toby's  check out your channels right link swapping the   data to link the description for you guys and  as always that's it and catch houses bye - bye

Channel: blackpenredpen

Views: 418,591

Keywords: sin(3 degrees), sin(pi/60), math for fun, special special triangle, sine of 18 degrees, angle difference formula, exact value of sin(3 degrees)

Id: GldJWGojFcs

Length: 33min 16sec (1996 seconds)

Published: Wed Jun 05 2019