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in the calculus playlist we came up with a rationale for Euler's formula or Euler's formula and I guess that video has gotten somewhat famous with me talking about Minds being blown and people not having emotion but just as a reminder Oilers formula is that e to the I theta and this just the formula by itself is pretty amazing but it becomes especially amazing when you substitute when you a substitute PI or tau for theta and if you don't know what tau is there are videos on that but or lers formula tells us that e to the I theta there's very good rationale for saying that this is going to be equal to this is going to be equal to cosine of theta plus and I'll just change the colors for fun plus I I sine of theta and so one thing I want to do is explore this a little bit can i express because of Euler's formula can I now express cosine of theta in terms of some combination of the other stuff it can I now express sine of theta in terms of some other combination of the other stuff well let's think about how we can isolate a cosine theta if this is e to the I theta let's see what e to the let's see would e to the negative I theta it would be negative I theta this would be equal to this would actually let me do it over here let me do it over here on the right hand side so e to the negative I theta e to the negative I theta this is the same thing as e to the I times negative theta which would be equal to by Euler's formula it would be equal to cosine of negative theta plus I sine of negative theta plus I sine of negative negative theta well cosine of negative theta this is the same thing as cosine of theta that's the same thing as cosine of theta and sine of negative theta is the same thing as negative sine of theta so this whole thing simplifies to negative I sine of theta so we can write e to the negative I theta we can write e do that pink color we can write e to the negative I theta as being equal to cosine of theta cosine of theta minus I sine of theta minus I sine of theta and now this is something we have something pretty interesting going on if we add these two expressions these two equality's right over here the left hand side the left so let's just add them up the left hand side we're going to be left with e to the I theta e to the I theta plus plus e to the negative I theta and on the right hand side these I sine Thetas are going to cancel out with each other the I sine theta is going to cancel out and we're gonna be left with two times cosine of theta two cosine theta and so now it's pretty straightforward to just isolate the cosine beta divide both sides by two divide both sides by two and we get something pretty interesting we get that cosine of theta is equal to this business ETI theta plus e to the negative I theta all of that all of that over two and we could do a similar type of thing to try to isolate let me rewrite it over so let me rewrite it this little discovery we made cosine of theta can be re-written as e to the I theta plus e to the negative I theta all of that over 2 so that's kind of interesting we can now Express cosine of theta in terms of a bunch of Exponential's right over here let's see if we could do the same thing for sine of theta so sine of theta let's imagine e to the I theta and let's try to do the other one you know there's always these weird kind of parallels between cosine and sine so let's just see what happens if we were to subtract this from that so let's try doing that if we were to take e to the I theta let me just rewrite it over again e to the I theta is equal to cosine of theta plus I sine of theta plus I sine of theta and from that let's subtract let's subtract e to the negative I theta well now we're going to subtract this so we're gonna get negative cosine theta negative cosine theta and then we're subtracting this so plus I sine theta so plus I plus I sine theta and so we are left with we are left with 1 we do the subtraction the left hand side we have e to the I theta e to the I theta minus e to the negative I theta minus e to the negative I theta is equal to this cancels out and we're left with 2 I sine theta 2 I times sine theta if we want to solve for sine theta we divide both sides by 2 I divide both sides by 2 I and we are going to be left with sine theta is equal to this business right over here so let's do this so sine theta so this is our other discovery sine theta can be written as e to the I theta minus e to the negative I theta all of that all of that over 2 I so there's something interesting about these structures I can write cosine of theta as the sum of these two Exponential's that are dealing with imaginary numbers sine of theta as the difference of these two exponential numbers dealing with imaginary numbers actually over here it's actually the average of the two functions you're dividing by 2 here we're dividing by 2i and if you didn't want if you don't like that I in the denominator you can multiply the numerator and the denominator by I this will become a negative you'll have the I in the numerator but either way you will have the same expression so this is all all I can say about this is this is just kind of interesting it's just kind of a fun exploration of mathematics but maybe there's something neat about the structure of these functions what happens if we were to take functions like this but if we were to essentially get rid of the eyes right over here they seem like there might be some kind of an analogue between cosine of theta and sine of theta and so let's define those functions and in the next few videos we'll explore them and see if they really are analogous in some bizarre strange and beautiful way to cosine of theta and sine theta so we're just gonna make new function definitions and they're just being inspired by these new ways that we discovered to write cosine of theta and sine theta and I'm going to call the first function I'm gonna call it Kosh and it really does come from hi Birbal ik cosine and we'll explore more why we call it hyperbolic cosine but let's just call it coach I'm just experimenting right over here and I will just say coach of let's say coach of X is equal to it's gonna be all of this it's going to be inspired by this but we're gonna get rid of all of the I so it's going to be e to the X plus e to the negative X all of that over to once again just inspired by this right over here getting rid of all of the eyes and then let's define another one hyperbolic sine so I'll call it cinch cinch for a hyperbolic sine sign I guess you could also do of X and I'm just gonna get rid of all of the eyes I'm just being inspired by this so e to the X minus e to the negative x all of that over 2 and in the next few videos we're going to start exploring this we're gonna start realizing that there is this strange bizarre parallel between these things and trigonometric for enter and our traditional trigonometric functions now these will show up in some of your mathematics but it is important these aren't as core especially in traditional mathematics as these two characters are here but it is a fun exploration