the most DISLIKED math notation
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- Опубликовано: 28 сен 2024
- The rules of exponents make sense. 3^-1=1/3 and x^-1=1/x but f^-1 doesn't mean 1/f
f^-f is one of the most problematic math notations or one of the most disliked math notations. f^-1 actually means the inverse of the function f. We use this notation a lot, especially for trigonometric functions. For example, tan^-1(x) means the inverse tangent, or we can also write it as arctan(x).
BUT!!! What exactly is tan^-2(x) supposed to mean?
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This video could be entitled “the reason for the existence of cotangent”
and of secant and cosecant
😂😂
Well, historically, the reasons for the existence of the cotangent are completely different, but yes.
Where do you see the cotangent???
@@HugoHabicht12 because tan^-2(x)=cot^2(x)
this is why normal people use arctan
No. Normal people use brackets.
Normally, one would use atan2(,) instead of arctan(), in order to avoid division by zero in its argument.
IDK, for me most common notation is "arctg", "arcsin" e.t.c.
@@AugustinSteven what like (tan(x)) to mean inverse tangent, or tan()(x), or is it tan^() (x)
@@AugustinSteven Hi Steve, I think that the primary problem is not with brackets.. problem is with naming of the function ; different functions (tan() and arctan()) using same name with adding few numeric symbols (^-1) which are in mathematics already defined for another purpose. Power symbol ^ has some previous definition, which is different. Writing tan^-1(x) for meaning inverse tangent function is VERY VERY inappropriate symbolic representation. Much better representation are also argtan(x), invtan(x) , atan(x) .... but I dont see a reason, why dont use name arctan(), which is strictly defined; and chance of misunderdstaning is really minimal. Have a nice day.
How about for inverse functions we just write it with the function name itself inverted? So not arctan() by rather, uɐʇ().
Wayne VanWeerthuizen
Yooooo how did you type that upside down?????
Btw, that reminds me of the omh and hmo : ))))
Too bad I am on phone and don’t know how to type those symbols up.
@@blackpenredpen mho*
-Chromium- yes
So given the famous Hamiltonian function H (X, ....) how would we write its inverse?
I prefer the arc notation, it clears up the confusion
i agree
But it's like he mentioned, traditionally it's with the negative1 superscript which was how I was taught and I'm completely fine with it. Personally, writing the arc is a bit much for me.
Jason Lu Yeah I get that, maybe they could shorten the arc part to just a or alpha so it's like atan(x) or αtan(x)
Jason Lu Except writing arctan on a computer is more manageable than tan^-1. Also, the arctan notation is inherently consistent and less confusing.
My solution is to never actually use the notation tan^2(x) unless I authentically mean tan[tan(x)]
in Russia we only use the arc notation. We also write "tg" instead of "tan". I guess you could call it our math dialect :D
same in Poland
Same in Peru.
And also ctg instead cot or cotan, cosec instead csc and sec intead sc
Same in croatia, tg and ctg, inverses are arctg and arcctg
Same in Ukraine
The main problem is the notation tan²(x) = (tan(x))².
With usual functional notation f²(x) = f(f(x)), so it would make more sense for tan²(x) = tan(tan(x)), which is consistent with tan^-1(x) = arctan(x).
But the prior was introduced because without brackets common expressions of tan x ² etc were ambiguous, probably due to laziness of not wanting to write brackets.
With this logic, tan -²(x) should equal arctan(arctan(x)).
f^2(x) = f(f(x)), but not for sine! sin^2(x) = (sin(x))^2. I don’t like these inconsistencies, it should be all or nothing with maths.
I agree. I say that sin^2(x) should be treated as different notation as sin(x)^2. Simple
Especially given sin^-1(x) can mean arcsin(x) or 1/sin(x)
If I designed the notation this is how I would do it:
sin^n(x) = sin(sin(...(x)...)) {n times} ∀n∈ℤ+
sin^0(x) = x
sin^-1(x) = arcsin(x)
sin^-n(x) = arcsin(arcsin(...(x)...)) {n times} ∀n∈ℤ+
sin(x)^p = (sin(x))^p ∀p∈ℝ
Of course, this only works if we *always* use brackets whenever we use the trig functions, but I don't see this as a problem as we already do so when we write f(x), and all of the notation I've just described is already in use when we talk about a general function f(x), e.g. f^2(x) = f(f(x)), f^-1(x) is the inverse function of f, f(x)^2 is (f(x))^2.
I think you're notation is pretty cool but, how useful is sin (sin(X))
And what about arcsin(arcsin(X))?
Well it's not necessarily about usefulness, but rather about consistency. This is how function composition notation works when we write f(x) so I don't see why it wouldn't work for other functions. This would apply to non-trig as well, e.g. ln^-1(x) would be e^x, and ln^-2(x) would be e^e^x, etc.
EDIT: and of course this prevents the confusion that we saw in the video.
Okay, so if f^2(x) is f(f(x)), what is f'2(x)?
@@zeeshanmehmood4522 if you mean f'^2(x) it's f'(f'(x)). If not I haven't heard of this notation before. If you mean differentiate it twice it's actually f''(x) or f^(2)(x). (2 should *always* be in a parentheses in order not to be confused with powers).
Hope that helped.
What @shayan moosavi said. For some reason RUclips didn't notify me when the responses happened and I only saw it now because I got a notification stating "Somebody liked your comment" 😐
Arc notation clears all confusion and it actually sounds really cool
I agree, but it's six letters long. I don't like that.
@@radadadadee you can write atan but be sure to specify it
@@Errenium Russian?
But how do you handle f^-1(x)?
@@BetaKeja Exactly. That is the question. This idea of "arc" notation only creates further inconsistency, as it only makes sense from trigonometric functions.
Another benefit to the arctan notation is that it is a _specific_ inverse (that is, with -π/2 < x < π/2), rather than just a generic inverse of tan. Same goes for arcsin, arccos, arcsec, arccsc and arccot, of course...
Fun fact about inverse hyperbolic functions by the way: the reason they use ar-, rather than arc-, is because 'ar' stands for 'area', since the resulting value tells you about the area between a hyperbola and a line from the origin to a point on the line, whereas 'arc' is of course refering to the arc length of a unit circle.
Funnily enough, if you type arcsin(1.5) in Wolfram Alpha, it gives you an answer (a complex number).
(+Lyri Metacurl) That’s because that’s mathematically correct.
I like how you always have something to share with us...shows you passion in maths...love your content man
Simphiwe Dlamini thank you!!!!!
In the grand, vast mathematical literature, (f^n)(x) almost always has referred to iterations of f, not multiplicative powers of f. Multiplicative powers are expressed as [f(x)]^n or more simply f(x)^n, since what you are repeatedly multiplying is the output of f given some input x, not f itself. tan^(-1) is should be the correct notation for inverses. The problem would be solved if people simply stopped writing (tan^2)(x) to mean tan(x)^2, because for any other function, (f^n)(x) and f(x)^n are most definitely not the same. Yes, it may be a tradition, but some traditions objectively should stop existing. This is one of them.
However, since it's probable that mathematicians will never stop committing the aberration of writing (tan^2)(x) to mean the "tangent of x " squared, I think it would be best that instead we used arctan as opposed to tan^(-1). As for tan^(-2), it would be best to use it to represent arctan(arctan(x)). Then at least some sort of rule could be established where negative exponents do represent iteration and the positive do not. This is much less ambiguous and much more consistent than simply using every exponent as multiplicative powers except for -1 to mean functional inverse. That is just deluded.
tan⁻²(x) should be tan⁻¹(tan⁻¹(x)) which should be inverse tangent of inverse tanget of x but it's actually interpreted like 1/(tan(x))²
tan⁻¹(x) should be inverse tangent of x and it is indeed defined that way.
tan⁰(x) should be x and it is indeed defined that way.
tan¹(x) should be tan(x) which should be tangent of x and it is indeed defined that way.
tan²(x) should be tan(tan(x)) which should be tangent of tangent of x but it's actually interpreted like (tan(x))²
The thing I hate is the fact that they apply regular function notation on every function except the trig ones and log and ln. WHY?? Is it that hard to write parentheses? How do you know if logxi means log(x)i or log(xi)?
addendum: So I actually typed tan⁰(x) in wolfram alpha and it spat out 1
@@xXJ4FARGAMERXx The answer is that people are lazy, and often fail to understand how to use mathematical notation correctly. Most people see no difference between writing (sin^2)(x) and writing sin(x)^2, even though they are notation representing different things.
@angelmendez-rivera351 True, indeed we often are lazy creatures as well as creatures of habit. However, in this case, I don't think it was due to laziness. It was more so that mathematicians in more classical times wrote with a certain style, a certain elegance, and with intention too. They saw the aesthetics of mathematics, not just at a conceptual level, but in its raw expression through notation. My point is that, when they wrote, as best as they could to avoid confusion and write with consistency, seeing that that was also part of the beauty as well as the utility of it all, to make the notation/machinery usable, you can't always have it perfect. But they relied on people reading their writings to have a certain level of mathematical maturity, that they will be able to "get it" without much hassle and move on, so long as it wasn't totally convoluted, bulky, awkward, ugly, etc.
@@daxramdac7194 I don't think what you have said addresses anything I have said at all, so as far as I can tell, this definitely still just boils down to laziness.
(tan x)^n and no more confusion. Tbh, I prefer this notation even with logarithms
Exactly. That or tan(x)^n, consistent with the fact that tan(x) = tan x is just some number y, and powers of numbers are written y^n. This also distinguishes it from (tan^n)(x), which should be reserved to denote n applications of the tangent function, which is consistent with how the notation is used in mathematics for functional iteration and operators, keeping in mind that operators are just a special type of function. Consistent, unambiguous, and concise. Finally someone who understands.
The inverse tangent should be known as Euler's Confusion.
XD
XD
YD
XD
ZD
tan²(x) should get redefined as tan(tan(x)), then the - exponent could still mean inverse: tan-²(x)= arctan(arctan(x)).
Or make a new term for the inverses, like tān(x), sīn(x), cōs(x)
There is no shorthand notation for tan(tan(x)), arctan(arctan(x)), etc. We just leave as this type of notation as the composition notation.
No need to redefine. It's already defined correctly in your Linear Algebra book. f²(x)=f(f(x))
oof
@@justabunga1 actually, no. The exponentiation of endomaps is well defined and largelly used in group theory.
Because I found the notation ^(-1) so confusing, I had decided to use a ~ above the function instead
I've seen tan^← notation (with an "exponentiated back arrow behind the function name"). To me, that is clearest. I would prefer it over arctan because it works on any function. Disadvantage is that it is uncommon.
But, to be fair, the standard f^-1 extends nicely to function composition f^2(x)=f(f(x)), so if anything using it as the square instead of (tan(x))^2 seems like the bad notation.
Why don't we always just use arctan instead of tan^-1?
Because we always tend to find shortcut ways
I prefer at(x)
In Russia we do. But we also use "tg(x)" instead of "tan(x)", so...
Because we always use the notation f^-1 for any function
You also write sqrt (x) as (x^2)^-1 ?
I actually hate the whole exponent, log and root notation more than this one. Ones a word, ones a symbol, ones a position, and they are all related. Really awful stuff.
Yeah your dp already told so.. 😅
I hate the "dx" thing in integrals, if you have an integral of d then it becomes
⌠1
│ ddd
⌡0
@@groszak1 Not a problem if you do it right: ∫𝒹·d𝒹
@@bob53135 what if both d and 𝒹 are used as variables?
@@groszak1Then you've got a confusing and ambiguous way to write variables (should be italic) and notations (should be non italic) and this is why you can't have nice things :)
Wait, do we still have the inverse tangent?
tan⁻²(x)=42
Tan ( tan( 42)) = x
Boi.
No Vivek
@@vivekmathur3514 Get to know interesting people around you. Swipe right to like, and swipe left to forget. Tantan is a fun and easy way to connect you with the people you like.
Ri Soo Keu eat wiener
BUT
NO VIEWS WHY???
tanuj kumar I have no idea....
@@blackpenredpen I think it's about the problem of counting from multiple inputs. Tom Scott made a video about this. Every time someone watches a video, it is supposed to add 1 to the views counter; the problem is, if two or more people watch the video at the same time, they see the same views count, thus they add 1 to the same number, instead of adding 1 to the original count and then 1 again to the new count. To solve this problem, they had to work out a system which sometimes lags and takes a while to output the actual count.
@Abbhinav Bharadwaj That is also true. But no need to be so rude.
@@TreniFS_ whom you are telling 🙄🙄
My guess is that tan^(-1) is explicitly defined as arctan in math programs because it's a common notation, but other negative numbers are just done how the program usually handles exponents.
Arc notation is much easier to understand. And in Russia we use tg and arctg instead of tan and arctan
In Portuguese speaking countries we use tg cotg as well
@@marcioamaral7511 I'm portuguese and I use tan, cos and sen (not sin) and both tan^-1 and arctan for inverse trigonometric functions
I think I will rebel and just be using overlined for its inverse, not that bad idea (though better than that disliked notation)
Then write 1000 arctan(x) and tan^-1(x) in 2 different papers. Compare how much ink to write them. You will know why inv functions exists
Next I'd like to see a video on how/why exponents are used the way they are for 2nd, 3rd, 4th ... derivatives.
The subscript for the derivative notation is a Leibniz notation (e.g. d^2y/dx^2).
It probably has to do with how nth derivatives can be written as a summation that involves binomial coefficients. And binomial coefficients are used when raising a sum of two terms to an exponent. So maybe it’s related to that?
teavea10 It’s probably when you think of d/dx as an operator and you are talking about how many times you are applying the operator
@@georgedoran9299 Yes, that is the correct explanation. This is why the D[y(t)] notation to refer to the derivative D applied to the function y is just better. Less confusing, less misleading, and easier to type and read.
@@angelmendez-rivera351 the problem is when the thing your differentiating is not y(t), just y.
5:16 turn on captions.
XD
When I went to school forty years ago, _arctan x_ meant all values for _x_ and _Arctan x_ meant only the principal value of _x_
[-π/2 < _x_ < π/2]. I still use the notation I learned and none of my students complain that it's different than what's in their textbooks... they know I'm old-school.
When I guessed the tan^-2(x), I thought it would be something like this: if tan^-1(x) is the inverse of tan(x) then tan^-2(x) might be tan(tan(y))=x and then solving for y. However I am glad the real meaning was simpler.
For inverse function notation in general, maybe we should change it to "f" arrow instead.
As soon as I knew about the arc notation I quickly abandoned the ^-1 notation. I really wish the ^-1 notation wasn’t used at all
Personally I don’t like the dot notation for derivatives in respect of time
Weighing in on the epic feud of Isaac Newton vs. Gottfried Wilhelm Leibniz?
In Classical Mechanics is a tradition and a sort of identity, because it's the Newton notation. But just in Classical Mechanics.
Iike the D notation of D'Alambert and Euler
Using the notation D[y(x)] to refer to "the derivative D applied to the function y" is just objectively the best notation. It is simple and concise to use, without weird symbols that cannot be typed, and is not notationally misleading like the Leibniz notation, and it lends itself to generalizations for multivariable calculus, differential geometry, and linear operator theory much better. It makes the study of ODEs that much more convenient, as well.
same with f²(x), it can mean (f×f)(x) or the second derivative of f(x)
Wikipedia and new text books lost parentheses y=sinx and that is confusing y=sin(x)^2, and y=sin(x^2). And sin^-1 it also new thing compare to old times where arcsine, means you try find angle of the arc using inverse sine function.
Totally agree!!!
Hold on, did dr. P just comment on my video????
I love how I aced Trig, thought I was taught to graph anything, & saw this in Calc1: ƒ(x) = x³ cos(x) 🤷🏻♂️
At 1st I thought I had the wrong textbook, we didn't have to graph it but still...the horror. (Yes, "Apocalypse Now" reference)
Unpopular opinion: instead of using arctan, tan^2 x should equal tan(tan x) instead of (tan x)^2
I think this notation would be more consistent as we have things like d^2(x) = d(d(x)) and not (dx) ^2. After all, we're writing the 2 in front of tan, and not the whoele expression tan x, so doesn't it make more sense to do tan twice instead of the whole expression twice?
Edit: I should also touch on tan^-2 x. Since we're writing in superscript, it would make sense for the power rules to apply. So I suggest that tan^-2 x = (tan^2)^-1 x = (tan tan)^-1 x = tan^-1 tan^-1 x
= (tan^-1)^2 x
= tan^-1 tan^-1 x
It works out mathematically, it's consistent, and it just makes so much more sense
I agree
Pretty sure that isn't unpopular at all.
People that write tan¯¹(x) for arctan(x) (and equivalents for the other trigonometrics) are the same people that write log(x) for ln(x). The second one is a pet peeve of mine; I thought we had already established that log(x) was for base 10 logarithm, don't create unnecessary confusion by also using it for base e!
6:06 i thought it would be "tan inverse of tan inverse X"
That’s like saying arctan(arctan(x)), so no shorthand notation for this.
@@justabunga1 Except that is the correct shorthand notation.
They should've just kept the reverse trig operators as being the arc-functions.
Hi I agree. Also in spherical coordinates phi and theta get swapped depending on which text book you are reading.
tan^2(x) sometimes has another meaning, namely composition: tan(tan(x)). I guess composition can be regarded as a multiplication when you don't have one. For example, consider linear maps f, g and h between two fixed vector spaces. Then we have fg+fh = f(g+h), where the "multiplication" is now composition, meaning the set of said maps is a ring, which is quite nice. That said, it can surely cause confusion, but I guess tan^2(x), tan(x^2) and tan(x)^2 represent all different meanings this kind of term can have.
I have a particular trigonometric notation. Firstly, I live in Brazil and in portuguese sine is written "seno" so sin(x) is written sen(x); cos(x) is cos(x) and tan(x) is tan(x), although I particularly write tg(x). The inverse functions are arcsen(x), arccos(x) and arctan(x) but I write as asen(x), acos(x) and atg(x). C´mon... much easier write atg(x) than arctan(x), isn´t it? It makes expressions much shorter.
Dear blackpenredpen: As a lifelong math, science and engineering enthusiast and practitioner, I would really like to commend you for the simply brilliant mathematical presentations in this series. THEY ARE RIVETING! Your command of mathematics and clarity of derivations are totally amazing. As a long-time course grader, I happily assign you an A+ for this series. Keep it going.
No views
473 likes
No views=473
Worst math notation, no is 473
Sorry, there was a mistake in the editor,,,, the correct equation is like this
F=curl4(gradient19)^119+256f(A,B)div(m)
i will call this the devil's notation
I only had to look at the thumbnail to know the majority of this video.
I know the struggle too well, having taught essentially the same lesson to students many times.
In the context of dynamical systems tan^(-2)(x) means arctan(aractan(x)), ie, the second iterate of the inverse of the tangent. The same is true for tan^2(x) = tan(tan(x)).
i think i read somewhere on wikipedia to use capital letters (like Sin^-1 (x) instead of sin^-1 (x) ) when denoting the inverse trigs
I thought tan ^-2 (x) would be arctanh(x) . :-)
I always use arctan (x) instead of
tan^-1(x) .
Loved the jazz exit.
i remember a limit question involving sin(x)*sin^-1(x) (and some other stuff), i really thought i could get away with just cancelling, but i clarified with the professor and sadly my worst fears were realized
No views but 500 likes... Yeah, well done, you should use the black pen for the easy counting..
OnlyTheBest
Magic!
Dope music at the end! And I've always really disliked inverse trig notation like that. Once I learned about the arc~ notation I never went back, I just calmly deal with it everytime I see it in books and lectures. As long as I know my notation is unambiguous I'm happy
PS I'm currently tutoring some middle school kids and 9th graders and this reminds me of the same confusion they have with order of operations. Mainly the ÷ symbol and parentheses. Parentheses are hard as hell to explain to someone who's not used to how they are used. It can mean so many things! And being precise and concise while not being too abstract about their use can be a real challenge. But that feeling of knowing they understand something they didn't before is priceless!
tan^(-1+0)(x)
tan^(-1*1)(x)
gives cot(x)
tan^(cos180)(x) gives something completely different
Really? Okay, who was the rebel that disliked?
Oh. The thing is -1 is thinking in inverse for the composition operation and the 2 is for the multiplication operation. Most times we get it from context but it actually IS wrong (the notation). But also... a good notation might be ugly. I like to overnotate when things aren't totally clear and simplify notation when there is no chance of missing.
I didn't get a notification for this video 🤔
I haven't gotten a view yet, but there are like over 200 likes...
@@blackpenredpenI noticed the same thing! Your ratio of views to likes is very good .. if it exists.
I don't like the notation.
Down the Rabbit Hole lol
Hi, Just a question, What is the inverse of this function: u = (e^(−2) * (2^x)) / x! || so, x = ????
I'm having a problem with the factorial part. some on SE suggested that the inverse is n = e exp(W(1/e * log(n!/√2π))) − 1/2 where (W) is the Lambert W-function.
BTW this is the CDF function of the Poisson distribution. thanks dude. =)
I have had so many math classes where we simply were not allowed to use arctan and I hated it so much. It wasted a lot of time trying to figure out notation rather than just doing clear math with arc notation.
/r /TeamArctan
yeah #'s are old now
I think its because in most cases arc functions arnt squared, ive never seen a problem where its like atan^2 but I have seen cot^2, so for ease of use they probably made it like that
The function x^-1 is its own inverse. So basically
(x^-1)^-1 = x^-1.
Can you think of a more confusing mathematical notation?
There are 2 notations that I hate more: Number 2: Minus sign used both for negative and for the subtraction operation, which forces to treat negative numbers in a different way than positive numbers. For example 2*a = 2*3 if a=3, but 2*(-3) if a= negative 3. If there was a negative indicator (or perhaps a negative AND a positive indicator) that is different from any operator, then it would be much simpler and better. Things like "plus negative 5" would make much more sense and it would be much easier to explain to small children in elementary school. Number 1: We teach that the * sign can be omitted, so 2a=2*a, or 2(3-1)=2*(3-1), or 1/3a=1/3*a (imagine that the / is the horizontal bar centered vertically with the a). And then 2 1/3 = 2+1/3 and not 2*1/3? What for? To save one plus sign? When we say 3 and 4 we write 3+4, I don't see why 2 and 1/3 cannot be 2+1/3. I hate mixed fractions notation.
Yes, mixed fraction sucks a lot. If you wanted to write the number as one number just put it as an improper fraction, namely 7/3
maybe tan^-2 (x) can be interpreated like arctan (arctan (x))
Dudurododo izi That IS objectively how it should be interpreted since (f^-1)(x) has always meant the inverse function of f and (f^2)(x) has always meant f[f(x)] in the literature. Don't know who came up with the terrible idea of (tan^2)(x) = [tan(x)]^2 and (sin^2)(x) = [sin(x)]^2, but it is a bad tradition either way.
But the inverse funcion of arctan is tan so I think it should mean tan(arctan(x))
@@valeriobertoncello1809 obviously tan(arctan(x)) =x
@@dudurododoizi8547 exactly
Not really but that actually means the same as (cot(x))^2. arctan(arctan(x)) has no shorthand notation for that or for repeated composition of functions.
My solution to this if there is any uncertainty is to distinguish: f^2(x)=f(f(x)) while f(x)^2=f(x)×f(x) and f^(2)(x)=f''(x). That's the convention I use anyway. Or for trig/hyp I try to avoid it by using arctan&cot/arctanh&coth
I have the way to solve all of our issues:
sec(x), cosec(x), cot(x)
I didn’t even know about arctan before now, but after seeing this it’s definitely what I’ll be using.
I always use arctan x, arcsin x, arccos x, etc. And also, I don’t write (cos x)^(-1) but sec x, etc
It is very confusing indeed. With two or more dimensional analysis you have something similar, it looks like a power but instead it's something like an ID. o_0
Before I really learned about functions, I thought that f²(x) means f(f(x)). For example sin²x is sin(sin x).
There’s no shorthand notation for this e.g. tan(tan(x)) is still tan(tan(x)).
Trigonometric functions are like that but not for all functions.
“YOU DONT HAVE THE INVERSE TANGENT ANYMORE”
In Ukraine we always use the arc notation
I don't /shrug
this is why my teachers avoided the negative 1 notation on trig functions: nobody knows if you mean the arc version or 1 over version.
I am a follower of you. After I pass class 12, I will make a math channel REDPENBLACKPEN.
Very original name
Is REDPENBLACKPEN the inverse channel of BLACKPENREDPEN? :D
@@sab1862 So would (BLACKPENREDPEN)^-1 be equivalent to REDPENBLACKPEN or 1/BLACKPENREDPEN?
i geuess the whole thing goes back to f^-1(x) means do the inverse function meaning the -1 always means do inverse but if it's anything else it means its a power
I dislike this notatation 😠
Inverse matrix notation is kind of strange too.
YOU _DONT_ HAVE THE INVERSE TANGENT ANYMORE
The great thing about WA is that it tells you what it interpreted your input as
'Hopefully we understand each other'
I used to write arctan(x) but after watching your videos and seeing when tan(x) and tan-1(x) cancel each other out I mean OH MY GOD IT'S SO SATISFYING !!!!!!
tan^-1(tan^1(x))=tan^0(x)=x (implying tan^1(x)=tan(x) and tan^0(x)=x
1815-1898 Iron Man tan^-1(tan^-1(x)) = x is most certainly not true.
@@angelmendez-rivera351 oops fixed it
Hahaha thanks
tan^-1(tan^2(x)) != tan(x)
For using with electronics such as wolfram alpha, use and take care of lots of parentheses to make sure you get the result you want, also check the out on wolfram alpha it will show you which function you typed in that it expected.
I dislike the notation but like the video xD
I'm so glad I'm not alone on this. The silliest thing is that if we wanted to use sub/superscripts to describe the inverse nature of a function, we could've just went with:
func_inv
And called it a day. Instead we have this ambiguity for ambiguity's sake. Arc notation is by far superior. If your equation has any variables named r, a, or c, you can simply capitalize them or use subscripts on them.
Arctangent would be so much clearer, because tan^-1(x) is the true inverse from opposite function compositions, however 1/(tan(x)) is technically the multiplicative inverse of tan (x). The nomenclature should keep these two "inverses" of the trig functions distinct, since it is all ready hard enough to distinguish from these each other without this notation catastrophy.
Arctan doesn't really solve the problem. That was just one example. You can't expect every inverse to have a special name.
What I actually dislike is the tan^2 x = (tan x)^2 notation. It's really common in many areas of math to indicate repeated application of a function with exponents. This is also the reason why -1 means the inverse function. E.g. if you have a function f(x), then f^2(x) = f(f(x)) (apply f twice), f^-2(x) = f^-1(f^-1(x)) ("unapply" f twice). If it were for me, I'd always write (tan x)^2 and reserve tan^2 x to mean what it's actually supposed to mean, which is tan(tan(x)). This removes any confusion and inconsistencies with the inverse function tan^-1.
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I personally always do ArcTan(x) when writing on paper. I capitalise my 't's because my lowercase 't's look too much like plus signs. Another notation I have saw before from a professor at my college is a small left-pointing arrow in the spot you would usually write your -1 for inverse functions. Might this be the right way to go?
Well this is because it checks for the special case of (-1) as inverse and the general case as exponentiation
In my opinion, tan^n x as a shorthand for (tan x)^n is the real problem here. If we dropped that, we could stick with the standard notation for inverse functions and iterated functions, i.e. f^-1(x) is applying the inverse of f and f^2(x) is f(f(x)). In that case, f^-2(x) would be the inverse of f applied twice, i.e. f^-1(f^-1(x)).
I guess the reason why we don't do this, is because there's little to no use of applying trigonometric functions iteratively. And also because mathematicians love to invent new notations to shorten expressions, even if it sometimes causes confusion.
Hi sir, blackpenredpen
To sum up the whole video, tan^n(x)=(tanx)^n, n=/=-1. If n=-1, use arctanx
^-1 means inverse of something. Is it after a function, it's the inverse function, is it after a number, it is the inverse number.
With bracket around, it meant inverse of the whole thing, without inverse of the last thing.
Inverse is only -1, not -2, so -2 makes no sense with inverse funktion, so it gets interpretated as something to the power of it.
So for me it is pretty clear, that tan x² = tan(x²) and tan x^(-1) = tan(x^(-1)).
@blackpenredpen can you do a video on solving (tanx)^-1 = tan^-1(x), ie cotx = arctanx?
It's weird to me that people don't *always* use parentheses when dealing with trig functions. When you have a function 'f', you say that y = f(x), not that y = f x !! So why, then, if trig functions are functions, too, does everyone not just use parentheses‽
I've not whilom been confused by that notation, but I definitely don't favour it: I utilise either atan(x) or arctan(x) to denote the inverse of these functions (referring to all to which this notation would apply). However, if I am denoting the inverse of an abstract or undefined function, I do denote it with some symbol and the ^(-1) before the list of parametres (i.e., π^(-1)(x, y, t)=...).
I've been watching your videos for a while now; they are quite helpful. I wonder, have you ever studied algebraic or differential geometry? (or any mathematical topics relevant to string theory)
I would definitely agree that
tan^(-2)(x)=(tan(x))^(-2)
:=1/(tan(x)tan(x))
is quite confusing (especially, if utilised out of context or in an inconsistent manner)
if we want to be consistent about notation. We choose one of two, EXACTLY one of two
1. tan^(-1) (x) as power notation, it means a function can have power, something like f^2 (x), which is the case for trig function (sin^2 (x), cos^2 (x), etc.), we have to use other notation for inverse function, preferably for any function.
2. tan^(-1) (x) as inverse function notation, it means if tan^(-1) (x) equals y, y equals tan x. but it also means that tan^2 (x), and something like that, cannot be use, or we will have this kind of nonsense otherwise, just like we have now.
So yeah, in our universe, we use tan^(-1) (x) as inverse function notation but use other superscript as power function. Both of the choice has different meaning. If we choose just one of them, it's completely fine, there's no conflict whatsoever. but mathematicians decide to use both
'YOU DON'T HAVE THE INVERSE TANGENT ANYMORE' - blackpenredpen