This is the definition we used in my first year calculus class! Or rather, we used this definition for cosine, and then I believe defined sine as the derivative of cosine. This "pure calculus" approach is quite interesting, allowing you to easily connect it back to the usual geometric definitions if you know them, but without having to introduce lots of geometry in order to be rigorous.
something is wrong, but I cannot identify the mistake. The derivative of sine is not sin' = sqrt(1-sin^2), that works only on the interval [-pi/2, pi/2]. Maybe that's it; sin and cos defined in this way as inverse functions are defined only on [-pi/2, pi/2]. You need to extend them further.
With some nice heuristics you were able to sneak up on the definitions of the mysterious sin and cos functions as well as the mysterious Pythagorean Theorem by discovering their properties, opening understanding like a flower, petal by petal. 🙂 Sin, cos, and e are so fundamental that nearly any new way of looking at them has value.
You could also define sine and cosine via functional equations, like f(x+y)=f(x)g(y)+f(y)g(x), g(x+y)=g(x)g(y)-f(x)f(y). I believe the only nice solutions to that system are f(x)=sin x and g(x)=sin x.
Very nice! I guess you can define pretty much any function this way, by first defining its inverse as an integral. This should be possible as long as the function's derivative can be expressed nicely in terms of the function itself. I've seen eˣ (or better, exp(x)) defined as the inverse of ln x, where ln x=∫₁ˣ1/t dt, but I haven't seen this method used to define the trig functions before. Just one thing though: the inverse of arcsin x is sin x restricted to [-π/2,π/2] and the inverse of arccos x is cos x restricted to [0, π], so a little extra work is needed to define sin x and cos x on ℝ.
This may be a dumb question, but do you just "have to know" the integral definitions initially? What I mean is: can those be derived without having to know what sin and cos are? I get that there's value in going in reverse order, but I still have that other question.
Oh my lord, thank you for saying alound sqrt(x^2) is abs(x). People failing to understand this is the source of a great many foolish online conversations. You know the ones.
I'm bothered by the use of y=Asin(x)+Bcos(x) as a solution to y"=-y as if it's a proven truth. I know it IS true, but how can its truth be proven before finishing the very definitions of sin(x) and cos(x)?
@@Loots1Of course there is. There's often multiple ways to define/approach objects in math, and often a lot of those definitions are more useful for specific applications/perspectives but are generally not great definitions to motivate an object. This is what andrey was evoking
@@themathyam And why should "motivating an object well" be the only criterion for the correctness of a definition? Let alone for different people the most effective way for them to motivate an object might be different, depending on their perspective, what they're interested in, their background knowledge, etc.
@19divide53 I didn't think I needed to qualify the fact this is a subjective thing. Nonetheless, the answer to your question is because usually when you are defining something, you want to give a definition that elucidates what your object actually is, or gives way to easy proofs of important properties. I did say it depends on what specific perspective/understanding you're trying to achieve, so of course some definitions are "better" for specific purposes. Nonetheless, there are definitely cases where some dedinitions are almost universally better than others. Usually a mathematician would say a definition is "morally correct" because it cuts into the essence of an object rather than defining it in a way that doesn't reflect its intrinsic properties, and this does happen a lot more often than you think. For example, mathematicians working with schemes usually have strong opinions about which is the "correct" perspective on schemes (locally ringed spaces, functor of points etc.)
ah sigh - this is an excellent insight into powers of equalities. It may be naively expressed - and there is no harm in that. Your thought processes arriving at what might be considered as a conjecture probably span a couple of centuries of math research. Well done! That, what you did, is, in my opinion, math in its most elegant form. There are other forms of math but really that is mechanical number crunching no? I mean - people usually do not become artists by coloring in or repeating work of previous masters. They learn by exploring. I am sure a phililolosopher or more observed something similar?
This seems to be circular. How can I know that the solutions to y''=-y are of the form y = Asin(x) + Bcos(x), if I am still defining sin and cos? It really seems to be begging the question
@@landsgevaer He only showed that those two functions satisfy y''=y, but not that *all* other solutions are a linear combination of them. But thanks to you, I now understand what he meant: 1 - thanks to the theory of differential equations, we know that y''=-y has solutions given by linear combinations of two linearly independent solutions; 2 - the sine and cosine defined in the video (as inverses of asin/acos in integral form) satisfy y''=-y; 3 - (and this is I think the part missing from/implicit in the video) the sine and cosine functions defined in the video *are linearly independent*, which follows easily from knowing that sin0=0, cos0=1, and sin(pi/2)=1 and cos(pi/2)=0. Thus, all solutions of y''=-y are linear combinations of such sine and cosine, and the rest follows. Thanks for your input!
@@TheEternalVortex42 The Wronskian is one tool to prove linear independence, but there are other methods. For example, knowing the values of the functions at two points can be enough, like in this case. In fact, assume there are are constants c,d such that c*sin + d*cos = 0 meaning that c*sin(x)+d*cos(x)=0 for ever real x. Recall that we know the values of sin and cos at x=0 and x=pi/2. If you substitute above, you discover that c=d=0, so sin and cos are linearly independent. To sum up, you don’t always need the Wronskian to show linear independence.
this is far more comprehensible than the "just memorize them lol" I got as explanation for doing calculus on trig functions in uni. shame it's 15 years too late.
Here's a simpler definition that is derived from the slope formula based on a linear relationship. Given two points A = (x1,y1) and B = (x2,y2) and using the slope formula m = (y2-y1)/(x2-x1) we can simplify it to m = deltaY/deltaX. If we consider the simple expression y = x or its function representation f(x) = x for a given linear equation; We can use the slope-intercept form of a line defined as y = mx+b Along with some basic arithmetic identity properties such as a+0 = a AND a * 1 = a To show that y = x is a linear combination where m = 1 and b = 0. We know that the 2D Cartesian XY plane has two axes that are orthogonal or perpendicular to each other. We also know that the line y=x or f(x) = x is a bisector of this plane in the 1st and 3rd quadrants. It is a diagonal line that bisects the perpendicularity. With that we also know that the angle of a Right Angle is 90 degrees or PI/2 radians. We can conclude that the angle between the line y = x and the +x-axis is in fact 90/2 = 45 degrees or PI/2/2 = PI/4 radians. We also know that the slope of this line is 1. What trigonometric function has a value of 1 when its angle argument is 45 degrees or PI/4 radians? It's quite simple it's the tangent function. We can now associate the tan(t) where t is the angle between the line and the +x-axis and the slope of the line as being equivalent. With that we can substitute tan(t) for our slope into the slope-intercept form of y = mx+b giving us y = tan(t)*x + b. From trig identities and substitutions we know that tan(t) = sin(t)/cos(t) and with that we can substitute again. We now have y = (sin(t)/cos(t))*x + b From the slope formula (y2-y1)/(x2-x1) simplified as deltaY / deltaX we can simply see the following definitions: deltaY = sin(t) deltaX = cos(t) Or from a difference between two points based on their horizontal and vertical rates of changes we can see (y2 - y1) = sin(t) (x2 - x1) = cos(t) Another observation from this is that the sine and cosine functions are orthogonal to each other. They have the same waveform (shape), the same periodicity, the same range and domain. The only two major differences between them is their initial starting values where x = 0 and they are sin(0) = 0 and cos(0) = 1 respectively. And the fact that the Sine function is an Odd function and the Cosine is an Even function. This also goes to show that 1 is Odd and 0 is Even and that 0 and 1 are Orthogonal or Perpendicular to Each Other. This can be proven through the definition of the Dot Product between vectors and their relationship with the Cosine Function. Yes PI is embedded within y = x. When something equals itself you end up going full circle!
If you define sin(t)=y2-y1, which means t is the input, but where do y1 and y2 come from? Given a value for t, which linear relationship do you choose? (One natural choice might be y=tan(t)x, but then what is tan(t) here? It cannot be sin(t)/cos(t) because those are not defined!)
@@19divide53 Consider two points on a given line in the form of y = mx+b where m is the slope of that line defined as (y2-y1)/(x2-x1). The two points are A = (x1,y1) and B = (x2,y2). The difference in the change in X which is the difference of the coordinate pairs of points in X as in deltaX = x2 - x1 which yields the vector deltaX is also cos(t) where t is the angle between that line defined by the two points and the +x-axis. The same can be said about the difference in the Y defined as deltaY = y2 - y1 which is also sin(t). The slope itself m as the ratio of the rate of change in height over the rate of change in width is defined as (y2-y1)/(x2-x1) = deltaY/deltaX = tan(t). We also know that tan(t) = sin(t)/cos(t) so the slope of the line m can also be defined as sin(t)/cos(t), thus making sin(t) = deltaY = (y2-y1) and cos(t) = deltaX = (x2-x1). This is true for any two points on any given line or linear equation. Take the line y = x. We can make a table of values for both x and y as follows: x | y ---- | ---- -1 | -1 0 | 0 1 | 1 We can take any two points on this line such as the points (3,3) and (5,5). From the slope formula we can calculate the slope: m = (5-3)/(5-3) = 1. This line has a 45 degree or PI/4 radian angle above the +x-axis within the 1st quadrant. The slope of this line is uniform as the rate of change or the displacement of the change in x with respect to the change in y with respect to the generated angle of the line above the +x-axis it proportionally equal. For every 1 we translate in height (Y) we also translate 1 in width (X). If we draw a horizontal line from the origin (0,0) to the point x on the +x-axis (x,0) this horizontal distance is cos(t). And from this point on the +x-axis at the point x we draw a vertical line up to the line gives us the point (x, y) where this vertical distance is sin(t). The ratio of the them is either tan(t) or cot(t) depending on which of the two are the numerator or the denominator of the fraction. If the fraction is deltaY/deltaX = sin(t)/cos(t) = tan(t) or, deltaX/deltaY = cos(t)/sin(t) = cot(t). If the algebra doesn't appear to make sense then draw it out geometrically. Here you'll end up creating a Right Triangle between the +x-axis and the line y = mx+b where the y-intercept b = 0 for all angles t that are between (0, 90). These are all acute angles. When theta = 0 or 360 degrees or 0 or 2PI radians, then the fraction of the slope m simply becomes 0/1 = 0. Try it with any two points along the +x-axis where the height y is always 0. Here you'll end up with sin(0)/cos(0) = 0/1 = 0. For any angle t that is greater than 0 and less than 45 degrees or PI/4 radians the slope will be fractional value less than 1. For an angle at 45 degrees the slope is exactly 1. And for any angle that is greater than 45 degrees or PI/4 radians upto 90 degrees or PI/2 radians the slope is greater than 1 and tends towards +infinity. At 90 degrees or PI/2 radians we have the following formula within the slope sin(90)/cos(90) = 1/0 = +/-infinity. Since we are evaluating this within the 1st quadrant only we can then state that this particular period of the tangent function approaches +infinity. If we were to evaluate the slope in the 3rd quadrant of tan(90) then it would tend towards -infinity. This is why the sine and cosine functions have a period of 2PI and the tangent function has a period of PI. This is also why there is a direct relationship between cos(t) and the dot product of two vectors divided by product of their magnitudes. This is also why the 0 vector is orthogonal to every other vector because 0 dot v = 0. And when the dot product results in 0, this means the angle between them is 90 degrees. You can try this with any two vectors. Consider the horizontal vector defined by the two points V0 = P1(5,2) - P0(3,2) and the vertical vector defined by the two points: V1 = P3(4,7) - P4(4,4). If we take the dot product between V0 and V1: V0 dot V1 will equal 0. This means that the two vectors are 90 degrees or PI/2 radians. We also know that cos(t) = (u dot v)/(||u|*|v|). It's because of these linear relationships of the trigonometric functions that are embedded within equality and assignment as in y = x as well as simple arithmetic as in 1+1 = 2 being a single case of a horizontally translated unit circle is why we are even able to do Calculus in the first place. Without the Secant and Tangent functions. We wouldn't be able to decompose or evaluate their slopes or their areas. The only thing that seperates Calculus from basic Algebra, Geometry and Trigonometry is also the process of applying the limits of a given function, equation, or expression. Everything within mathematics is all built off of this. You have an initial value (x) and you are applying some operation onto its operand yielding some result that is a type of transformation. Sure not all transformations are linear but most transformations can be decomposed to linear transformations. You find this within higher levels of mathematics such as Linear Algebra and Vector Calculus continued...
... continued As an analogy it is similar in nature to electronic engineering and computations within digital circuitry. We start off with basic components to construct our basic logic gates from various components such as transistors, resistors, capacitors, etc... and from there we abstract away from them and use the basic logic gates such as: AND, OR, NOT, NAND, NOR, BUFFER, XOR, & XNOR and with them we can built many various devices. We can then extrapolate away from those devices (IC - Integrated Circuits) and build more complex devices such as ALU (Arithmetic Logic Units), Memory Modules, Encoders or Decoders but in terms of logic they are typically called multiplexers, etc... And from them we can configure them in such a way to create state machines either Mealy or Moore. Furthermore we can extrapolate even more and begin to combine complex units with a set of varying buses, control logics, etc... and we have ourselves a Turing Complete CPU. It's no different in the layers of abstraction. They're all built off of a simple concept. They are built off of fundamental principles. Within mathematics even before addition or any kind of transformation begins, the simplest of all fundamentals are identities. Such as y = x. There is no change based on the given operation being applied. So for brevity sake we can say that y = x, a + 0, a * 1, a ^ 1 are all within the same equivalence class. There is no change or displacement taking place. This is also analogous to the 0 vector within Vector arithmetic as in that the 0 vector is arbitrary and has no linear relationship to any other vector. In other words, the 0 vector is 0 dimensional and it is a point of rotation, reflection, and symmetry. It's the starting point. I didn't mention the unit vector because the unit vector has a magnitude of 1 and it is displaced by an arbitrary unit of measure. When we look at the definition of a vector by the addition or subtraction of 2 points. This becomes clear. Taking an arbitrary point P(x,y) and adding or subtracting the 0 vector to it will result in no change to P in either the x or y direction. P remains unchanged. The ability to add, enumerate, calculate, compute is the ability to apply a given transformation. That's what mathematics is for the most part. Sure there are some other fields of math that are a bit more abstract such as number theory, game theory, etc... but even they still rely on the basic principles. Everything within all fields of mathematics are related. Just as how everything within the field of Physics in some way or another is related. From a philosophical perspective one can deduce that In order to perform mathematics one has to apply physics due to the fact that there is a transformation or a translation being performed since Physics itself is the study of Motion. Things that move are being translated or transformed. And quite conversely one can also deduce that one can not perform or understand physics without the fundamental understanding of mathematics. They both go hand-in-hand so to speak. You can learn about math without knowing physics sure (pure abstract theory) but to try and learn physics without first knowing math is kind of a moot point since Physics other than being the Study of Moving Objects in another general sense is (Applied Mathematics). Stand in place and rotate around 360 degrees. You location remained constant but your orientation did change with respect to time until you ended up back to where you started. Overall, there was no change even though you went full circle. This entire single revolution is also called a cycle and it has a given period of 2PI. And without knowing it you also generated an arbitrary unit circle with a radius of an unknown length where you were the center of that circle. You alone acted as a rotating 0 vector. Now if you stretched out your arm and did the same where a stick drew a line on the ground as you rotated in place then the length of your arm would be that arbitrary unit vector and we could say that your arm length is of 1 unit. Thus you generated the unit circle where you are the center of that circle and the length of your arm being 1 the radius is a length extending out from you. Thus you just performed the addition of 1+1 yielding 2. The radius of the circle is 1 and its diameter is 2. I suspect that one of the major things that many people tend not to perceive about 1+1 = 2 generating the unit circle is that they are thinking of it in a 1D linear fashion. The tend to forget or overlook the fact that Linearity can be applied to any higher dimensionality. 1+1 = 2 isn't just 1D mathematics, it's also 2D. How? We consider the fact that 1+1 = 2 is also 1*2 = 2. Here we have multiplication which by definition is an extension of addition in that it is repeated addition. When something is repeated it has a period to it. In other words it is cyclical. Sure addition is a linear transformation of translation from a 1D perspective. However, multiplication inherits that 1D linear transformation of translation as well but it also exhibits other types of transformations that can not exist in 1D space and require at least 2 degrees of freedom or more. The 1D aspect of transformation of multiplication is scaling or skewing. The 2D transformation of multiplication is rotation and with that rotation is where angles are generated and from that we are able to define area. Consider the following polynomial: f(x) = x^2. If we consider all values of x to be greater than 0. Then this portion of the parabola is a representation of all the corresponding geometrical squares where x is the length of the side and f(x) is its generated internal area. So even by simple addition of 1+1 = 2. We can also see that this is 1*2 = 2 and this would generate the area of a rectangle (not a square) whose side lengths are 1 and 2 respectively and the Area is 2 units^2. These two expressions are or can be within the same equivalence class. Also the expression of 1+1 = 2 also has a radial unit area of PI units^2. Because the Area of the Unit Circle is PI * r^2 and r = 1 and 1^2 = 1 and PI * 1 = PI. So yes PI is embedded within y = x, and 1+1 = 2 and so are the rest of the trigonometric functions and all other polynomials and exponential forms. This is why Euler's Formula and Identity work! This is why Fourier Transforms are so powerful. I didn't even get into complex numbers. Yet they are just an extension to the set of all real numbers as they are Orthogonal to the Real numbers. They are perpendicular to the Reals. If we get into number and set theory. I don't know if this is taught or not, but I would easily make the claim that the Empty, Null, or Zero Sets are Perpendicular or Orthogonal to the Full or Complete Set. Why? Simply because 0 and 1 are orthogonal to each other. The 0 vector is orthogonal to every other vector. It all has to do with the properties of points, vectors, lines, curves and the relationships between them. "Pattern Recognition" is one of the major aspects of mathematics.
The three most common phrases here: (1) a good place to stop (of course), (2) let's introduce some notation, and (3) we're going to use a trick.
I like it when he adds 0 and multiplies by 1.
"Let's look at if..."
“This motivates us …”
"Let's put this on the next board.
Ok, now..."
The parallels from these integral representations are exactly what have been helping me to understand Lemniscate elliptic functions better
This is the definition we used in my first year calculus class! Or rather, we used this definition for cosine, and then I believe defined sine as the derivative of cosine. This "pure calculus" approach is quite interesting, allowing you to easily connect it back to the usual geometric definitions if you know them, but without having to introduce lots of geometry in order to be rigorous.
something is wrong, but I cannot identify the mistake. The derivative of sine is not sin' = sqrt(1-sin^2), that works only on the interval [-pi/2, pi/2]. Maybe that's it; sin and cos defined in this way as inverse functions are defined only on [-pi/2, pi/2]. You need to extend them further.
With some nice heuristics you were able to sneak up on the definitions of the mysterious sin and cos functions as well as the mysterious Pythagorean Theorem by discovering their properties, opening understanding like a flower, petal by petal. 🙂 Sin, cos, and e are so fundamental that nearly any new way of looking at them has value.
Amazing approach :)
You could also define sine and cosine via functional equations, like
f(x+y)=f(x)g(y)+f(y)g(x),
g(x+y)=g(x)g(y)-f(x)f(y).
I believe the only nice solutions to that system are f(x)=sin x and g(x)=sin x.
Me during exam: let's first try fx=gx=0
@@satyam-isical Oh, oops... I forgot about those.
Very nice! I guess you can define pretty much any function this way, by first defining its inverse as an integral. This should be possible as long as the function's derivative can be expressed nicely in terms of the function itself.
I've seen eˣ (or better, exp(x)) defined as the inverse of ln x, where ln x=∫₁ˣ1/t dt, but I haven't seen this method used to define the trig functions before.
Just one thing though: the inverse of arcsin x is sin x restricted to [-π/2,π/2] and the inverse of arccos x is cos x restricted to [0, π], so a little extra work is needed to define sin x and cos x on ℝ.
This may be a dumb question, but do you just "have to know" the integral definitions initially? What I mean is: can those be derived without having to know what sin and cos are?
I get that there's value in going in reverse order, but I still have that other question.
Michael Penn! Given y=f(x), we say f is continuous at x=a if for each €>0, there exists §>0 such that
|x-a|
Yes, you can! Note that € is arbitrary. So saying less than or equal to an € is same as saying less than 2€ for example
How can we use that Asin(x)+Bcos(x) is a general solution for the differential equation without knowing what sin and cos are yet?
Oh my lord, thank you for saying alound sqrt(x^2) is abs(x). People failing to understand this is the source of a great many foolish online conversations. You know the ones.
16:54
why is the integral from 0 to 1 on 5:30 equivalent to pi/4?
Such an elegant definition...
It kind of reminds me as how you define elliptic functions as inverses of elliptic integrals.
Lorentz contraction has entered the chat.
I'm bothered by the use of y=Asin(x)+Bcos(x) as a solution to y"=-y as if it's a proven truth. I know it IS true, but how can its truth be proven before finishing the very definitions of sin(x) and cos(x)?
this is the most correct definition; with this approach, the elliptic sine would be a familiar elementary function for us
"most correct" this is a nonsensical term, something is either correct or its not, there is no such thing as most correct
@@Loots1Of course there is. There's often multiple ways to define/approach objects in math, and often a lot of those definitions are more useful for specific applications/perspectives but are generally not great definitions to motivate an object. This is what andrey was evoking
@@themathyam And why should "motivating an object well" be the only criterion for the correctness of a definition? Let alone for different people the most effective way for them to motivate an object might be different, depending on their perspective, what they're interested in, their background knowledge, etc.
@19divide53 I didn't think I needed to qualify the fact this is a subjective thing. Nonetheless, the answer to your question is because usually when you are defining something, you want to give a definition that elucidates what your object actually is, or gives way to easy proofs of important properties.
I did say it depends on what specific perspective/understanding you're trying to achieve, so of course some definitions are "better" for specific purposes. Nonetheless, there are definitely cases where some dedinitions are almost universally better than others.
Usually a mathematician would say a definition is "morally correct" because it cuts into the essence of an object rather than defining it in a way that doesn't reflect its intrinsic properties, and this does happen a lot more often than you think.
For example, mathematicians working with schemes usually have strong opinions about which is the "correct" perspective on schemes (locally ringed spaces, functor of points etc.)
ah sigh - this is an excellent insight into powers of equalities. It may be naively expressed - and there is no harm in that.
Your thought processes arriving at what might be considered as a conjecture probably span a couple of centuries of math research.
Well done!
That, what you did, is, in my opinion, math in its most elegant form. There are other forms of math but really that is mechanical number crunching no?
I mean - people usually do not become artists by coloring in or repeating work of previous masters. They learn by exploring. I am sure a phililolosopher or more observed something similar?
This seems to be circular. How can I know that the solutions to y''=-y are of the form y = Asin(x) + Bcos(x), if I am still defining sin and cos? It really seems to be begging the question
He proved it. He showed that if you define sin/cos as the inverses of asin/acos as defined by those integrals, that those functions satisfy y"=-y.
@@landsgevaer I think what you are saying is: to prove it, you assume it is the case (and then find out if it is or is not ... and in this case, is).
@@landsgevaer He only showed that those two functions satisfy y''=y, but not that *all* other solutions are a linear combination of them.
But thanks to you, I now understand what he meant:
1 - thanks to the theory of differential equations, we know that y''=-y has solutions given by linear combinations of two linearly independent solutions;
2 - the sine and cosine defined in the video (as inverses of asin/acos in integral form) satisfy y''=-y;
3 - (and this is I think the part missing from/implicit in the video) the sine and cosine functions defined in the video *are linearly independent*, which follows easily from knowing that sin0=0, cos0=1, and sin(pi/2)=1 and cos(pi/2)=0.
Thus, all solutions of y''=-y are linear combinations of such sine and cosine, and the rest follows.
Thanks for your input!
I suppose you need to show that the Wronskian isn't 0 so you know those encompass all the solutions
@@TheEternalVortex42 The Wronskian is one tool to prove linear independence, but there are other methods. For example, knowing the values of the functions at two points can be enough, like in this case.
In fact, assume there are are constants c,d such that
c*sin + d*cos = 0
meaning that
c*sin(x)+d*cos(x)=0
for ever real x.
Recall that we know the values of sin and cos at x=0 and x=pi/2. If you substitute above, you discover that c=d=0, so sin and cos are linearly independent.
To sum up, you don’t always need the Wronskian to show linear independence.
this is far more comprehensible than the "just memorize them lol" I got as explanation for doing calculus on trig functions in uni. shame it's 15 years too late.
I feel like everything would be simpler if you started with arcsin^2 and arccos^2
Oh Pythagoras. Finally, at 13:56 you square both sides. Wouldn't clearing the sqrt at the beginning be easier?
I'm still working on the basic definitions.
(arccos(x))' = -1/sqrt(1-x²) (to derive this, begin with: arccosx = y => cosy = x => -sinydy = dx => y' = dy/x = -1/siny = -1/sqrt(1-cos²y) = -1/sqrt(1-x²)).
Then: ∫from x to 1: 1/sqrt(1-x²)dx = -arccos(1) + arccos(x), but arccos(1)=0, so arccos(x) = ∫from x to 1: 1/sqrt(1-x²)dx
I found exp(x) = inverse(integral 1/x)
and that's a good place to stop
Here's a simpler definition that is derived from the slope formula based on a linear relationship.
Given two points A = (x1,y1) and B = (x2,y2) and using the slope formula m = (y2-y1)/(x2-x1) we can simplify it to m = deltaY/deltaX.
If we consider the simple expression y = x or its function representation f(x) = x for a given linear equation;
We can use the slope-intercept form of a line defined as y = mx+b
Along with some basic arithmetic identity properties such as a+0 = a AND a * 1 = a
To show that y = x is a linear combination where m = 1 and b = 0.
We know that the 2D Cartesian XY plane has two axes that are orthogonal or perpendicular to each other.
We also know that the line y=x or f(x) = x is a bisector of this plane in the 1st and 3rd quadrants. It is a diagonal line that bisects the perpendicularity.
With that we also know that the angle of a Right Angle is 90 degrees or PI/2 radians.
We can conclude that the angle between the line y = x and the +x-axis is in fact 90/2 = 45 degrees or PI/2/2 = PI/4 radians.
We also know that the slope of this line is 1.
What trigonometric function has a value of 1 when its angle argument is 45 degrees or PI/4 radians? It's quite simple it's the tangent function.
We can now associate the tan(t) where t is the angle between the line and the +x-axis and the slope of the line as being equivalent.
With that we can substitute tan(t) for our slope into the slope-intercept form of y = mx+b giving us y = tan(t)*x + b.
From trig identities and substitutions we know that tan(t) = sin(t)/cos(t) and with that we can substitute again.
We now have y = (sin(t)/cos(t))*x + b
From the slope formula (y2-y1)/(x2-x1) simplified as deltaY / deltaX we can simply see the following definitions:
deltaY = sin(t)
deltaX = cos(t)
Or from a difference between two points based on their horizontal and vertical rates of changes we can see
(y2 - y1) = sin(t)
(x2 - x1) = cos(t)
Another observation from this is that the sine and cosine functions are orthogonal to each other. They have the same waveform (shape), the same periodicity, the same range and domain. The only two major differences between them is their initial starting values where x = 0 and they are sin(0) = 0 and cos(0) = 1 respectively. And the fact that the Sine function is an Odd function and the Cosine is an Even function. This also goes to show that 1 is Odd and 0 is Even and that 0 and 1 are Orthogonal or Perpendicular to Each Other.
This can be proven through the definition of the Dot Product between vectors and their relationship with the Cosine Function.
Yes PI is embedded within y = x. When something equals itself you end up going full circle!
If you define sin(t)=y2-y1, which means t is the input, but where do y1 and y2 come from? Given a value for t, which linear relationship do you choose? (One natural choice might be y=tan(t)x, but then what is tan(t) here? It cannot be sin(t)/cos(t) because those are not defined!)
@@19divide53 Consider two points on a given line in the form of y = mx+b where m is the slope of that line defined as (y2-y1)/(x2-x1). The two points are A = (x1,y1) and B = (x2,y2). The difference in the change in X which is the difference of the coordinate pairs of points in X as in deltaX = x2 - x1 which yields the vector deltaX is also cos(t) where t is the angle between that line defined by the two points and the +x-axis. The same can be said about the difference in the Y defined as deltaY = y2 - y1 which is also sin(t). The slope itself m as the ratio of the rate of change in height over the rate of change in width is defined as (y2-y1)/(x2-x1) = deltaY/deltaX = tan(t). We also know that tan(t) = sin(t)/cos(t) so the slope of the line m can also be defined as sin(t)/cos(t), thus making sin(t) = deltaY = (y2-y1) and cos(t) = deltaX = (x2-x1). This is true for any two points on any given line or linear equation.
Take the line y = x. We can make a table of values for both x and y as follows:
x | y
---- | ----
-1 | -1
0 | 0
1 | 1
We can take any two points on this line such as the points (3,3) and (5,5).
From the slope formula we can calculate the slope: m = (5-3)/(5-3) = 1. This line has a 45 degree or PI/4 radian angle above the +x-axis within the 1st quadrant. The slope of this line is uniform as the rate of change or the displacement of the change in x with respect to the change in y with respect to the generated angle of the line above the +x-axis it proportionally equal.
For every 1 we translate in height (Y) we also translate 1 in width (X).
If we draw a horizontal line from the origin (0,0) to the point x on the +x-axis (x,0) this horizontal distance is cos(t). And from this point on the +x-axis at the point x we draw a vertical line up to the line gives us the point (x, y) where this vertical distance is sin(t). The ratio of the them is either tan(t) or cot(t) depending on which of the two are the numerator or the denominator of the fraction. If the fraction is deltaY/deltaX = sin(t)/cos(t) = tan(t) or, deltaX/deltaY = cos(t)/sin(t) = cot(t).
If the algebra doesn't appear to make sense then draw it out geometrically. Here you'll end up creating a Right Triangle between the +x-axis and the line y = mx+b where the y-intercept b = 0 for all angles t that are between (0, 90). These are all acute angles. When theta = 0 or 360 degrees or 0 or 2PI radians, then the fraction of the slope m simply becomes 0/1 = 0. Try it with any two points along the +x-axis where the height y is always 0. Here you'll end up with sin(0)/cos(0) = 0/1 = 0.
For any angle t that is greater than 0 and less than 45 degrees or PI/4 radians the slope will be fractional value less than 1. For an angle at 45 degrees the slope is exactly 1. And for any angle that is greater than 45 degrees or PI/4 radians upto 90 degrees or PI/2 radians the slope is greater than 1 and tends towards +infinity. At 90 degrees or PI/2 radians we have the following formula within the slope sin(90)/cos(90) = 1/0 = +/-infinity. Since we are evaluating this within the 1st quadrant only we can then state that this particular period of the tangent function approaches +infinity. If we were to evaluate the slope in the 3rd quadrant of tan(90) then it would tend towards -infinity.
This is why the sine and cosine functions have a period of 2PI and the tangent function has a period of PI. This is also why there is a direct relationship between cos(t) and the dot product of two vectors divided by product of their magnitudes. This is also why the 0 vector is orthogonal to every other vector because 0 dot v = 0. And when the dot product results in 0, this means the angle between them is 90 degrees. You can try this with any two vectors.
Consider the horizontal vector defined by the two points V0 = P1(5,2) - P0(3,2) and the vertical vector defined by the two points: V1 = P3(4,7) - P4(4,4). If we take the dot product between V0 and V1: V0 dot V1 will equal 0. This means that the two vectors are 90 degrees or PI/2 radians. We also know that cos(t) = (u dot v)/(||u|*|v|).
It's because of these linear relationships of the trigonometric functions that are embedded within equality and assignment as in y = x as well as simple arithmetic as in 1+1 = 2 being a single case of a horizontally translated unit circle is why we are even able to do Calculus in the first place. Without the Secant and Tangent functions. We wouldn't be able to decompose or evaluate their slopes or their areas. The only thing that seperates Calculus from basic Algebra, Geometry and Trigonometry is also the process of applying the limits of a given function, equation, or expression.
Everything within mathematics is all built off of this. You have an initial value (x) and you are applying some operation onto its operand yielding some result that is a type of transformation. Sure not all transformations are linear but most transformations can be decomposed to linear transformations. You find this within higher levels of mathematics such as Linear Algebra and Vector Calculus
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As an analogy it is similar in nature to electronic engineering and computations within digital circuitry. We start off with basic components to construct our basic logic gates from various components such as transistors, resistors, capacitors, etc... and from there we abstract away from them and use the basic logic gates such as: AND, OR, NOT, NAND, NOR, BUFFER, XOR, & XNOR and with them we can built many various devices. We can then extrapolate away from those devices (IC - Integrated Circuits) and build more complex devices such as ALU (Arithmetic Logic Units), Memory Modules, Encoders or Decoders but in terms of logic they are typically called multiplexers, etc... And from them we can configure them in such a way to create state machines either Mealy or Moore. Furthermore we can extrapolate even more and begin to combine complex units with a set of varying buses, control logics, etc... and we have ourselves a Turing Complete CPU. It's no different in the layers of abstraction. They're all built off of a simple concept. They are built off of fundamental principles.
Within mathematics even before addition or any kind of transformation begins, the simplest of all fundamentals are identities. Such as y = x. There is no change based on the given operation being applied. So for brevity sake we can say that y = x, a + 0, a * 1, a ^ 1 are all within the same equivalence class. There is no change or displacement taking place. This is also analogous to the 0 vector within Vector arithmetic as in that the 0 vector is arbitrary and has no linear relationship to any other vector. In other words, the 0 vector is 0 dimensional and it is a point of rotation, reflection, and symmetry. It's the starting point. I didn't mention the unit vector because the unit vector has a magnitude of 1 and it is displaced by an arbitrary unit of measure.
When we look at the definition of a vector by the addition or subtraction of 2 points. This becomes clear. Taking an arbitrary point P(x,y) and adding or subtracting the 0 vector to it will result in no change to P in either the x or y direction. P remains unchanged.
The ability to add, enumerate, calculate, compute is the ability to apply a given transformation. That's what mathematics is for the most part. Sure there are some other fields of math that are a bit more abstract such as number theory, game theory, etc... but even they still rely on the basic principles. Everything within all fields of mathematics are related. Just as how everything within the field of Physics in some way or another is related.
From a philosophical perspective one can deduce that In order to perform mathematics one has to apply physics due to the fact that there is a transformation or a translation being performed since Physics itself is the study of Motion. Things that move are being translated or transformed. And quite conversely one can also deduce that one can not perform or understand physics without the fundamental understanding of mathematics. They both go hand-in-hand so to speak. You can learn about math without knowing physics sure (pure abstract theory) but to try and learn physics without first knowing math is kind of a moot point since Physics other than being the Study of Moving Objects in another general sense is (Applied Mathematics).
Stand in place and rotate around 360 degrees. You location remained constant but your orientation did change with respect to time until you ended up back to where you started. Overall, there was no change even though you went full circle. This entire single revolution is also called a cycle and it has a given period of 2PI. And without knowing it you also generated an arbitrary unit circle with a radius of an unknown length where you were the center of that circle. You alone acted as a rotating 0 vector. Now if you stretched out your arm and did the same where a stick drew a line on the ground as you rotated in place then the length of your arm would be that arbitrary unit vector and we could say that your arm length is of 1 unit. Thus you generated the unit circle where you are the center of that circle and the length of your arm being 1 the radius is a length extending out from you. Thus you just performed the addition of 1+1 yielding 2. The radius of the circle is 1 and its diameter is 2.
I suspect that one of the major things that many people tend not to perceive about 1+1 = 2 generating the unit circle is that they are thinking of it in a 1D linear fashion. The tend to forget or overlook the fact that Linearity can be applied to any higher dimensionality. 1+1 = 2 isn't just 1D mathematics, it's also 2D. How?
We consider the fact that 1+1 = 2 is also 1*2 = 2. Here we have multiplication which by definition is an extension of addition in that it is repeated addition. When something is repeated it has a period to it. In other words it is cyclical. Sure addition is a linear transformation of translation from a 1D perspective. However, multiplication inherits that 1D linear transformation of translation as well but it also exhibits other types of transformations that can not exist in 1D space and require at least 2 degrees of freedom or more. The 1D aspect of transformation of multiplication is scaling or skewing. The 2D transformation of multiplication is rotation and with that rotation is where angles are generated and from that we are able to define area. Consider the following polynomial: f(x) = x^2. If we consider all values of x to be greater than 0. Then this portion of the parabola is a representation of all the corresponding geometrical squares where x is the length of the side and f(x) is its generated internal area.
So even by simple addition of 1+1 = 2. We can also see that this is 1*2 = 2 and this would generate the area of a rectangle (not a square) whose side lengths are 1 and 2 respectively and the Area is 2 units^2. These two expressions are or can be within the same equivalence class. Also the expression of 1+1 = 2 also has a radial unit area of PI units^2. Because the Area of the Unit Circle is PI * r^2 and r = 1 and 1^2 = 1 and PI * 1 = PI. So yes PI is embedded within y = x, and 1+1 = 2 and so are the rest of the trigonometric functions and all other polynomials and exponential forms.
This is why Euler's Formula and Identity work! This is why Fourier Transforms are so powerful. I didn't even get into complex numbers. Yet they are just an extension to the set of all real numbers as they are Orthogonal to the Real numbers. They are perpendicular to the Reals.
If we get into number and set theory. I don't know if this is taught or not, but I would easily make the claim that the Empty, Null, or Zero Sets are Perpendicular or Orthogonal to the Full or Complete Set. Why? Simply because 0 and 1 are orthogonal to each other. The 0 vector is orthogonal to every other vector. It all has to do with the properties of points, vectors, lines, curves and the relationships between them. "Pattern Recognition" is one of the major aspects of mathematics.
Ah the power of equalities knows no bounds. Or does it?
What the actual fuck
Early