@@suspendedsuplexchannel1000 they always have the same length so you can define it both ways. There's no "true" way to define things and this two ways to define tan can both be useful depending on the context. Where I came from we define tan exactly like that. It's actually very useful for rough mental estimation of tan value at any given angle, I would argue it's more useful for this then a standard definition in the West. In the end it doesn't matter, math is not about things themselves but about connections between those things. Or as Poincare sad it, "Mathematics is the art of giving the same name to different things".
If you are wondering what the cosine of a side of a spherical triangle means, referring to the spherical triangle he was drawing on the board: The sides of the triangle are arcs of great circles (geodesics) on the sphere. A great circle is a circle on the sphere, whose center is the center of the sphere. For example, if the Earth were exactly a sphere, all meridians of longitude would be great circles, and so would the equator. This means the sides of the triangle are arcs intercepted by central angles of this unit sphere. Since the sphere is of radius 1, the radian measure of these central angles is precisely the lengths of the arcs they intercept. So that is why it makes sense to talk about the cosine of a "side", rather than an angle, of the spherical triangle.
Transportation engineer here: The first time I remember seeing exsec and versine was while studying sophomore geomatics. Until recently they were both still used when laying out horizontal roadway curves by transport surveyors (commonly called "staking the right-of-way/centerline"). Reason? You can't shoot "through" an arc, but you can through the chord(s). The distance from the chord to the midpoint of the arc is a multiple of the versine. Likewise, you may have only one means to reach the point where the highway tangents would meet (called the P.I.), so to get back to the curve, you need the exsec. The actual on-the-ground distances are called the middle ordinate and external distance respectively. Of course, the angle in question is equals (or is twice depending on reference) the angle of deflection (the direction you will be driving/riding minus the direction you previously were driving/riding), and we (the roadway engineers) provide this in our submittal packets to the surveyor and the contractor. The funny thing is that, with the advent of differential GPS, curves can be laid out much more precisely without worrying about where the chords, middle ordinates, and external distances are, so basically nobody even knows how to use exsec and versine anymore.
the application of pythagoras' gives you the absolute value of the cord function, so |cord(t)|=2|sin t/2|. You can define it negative for negative angles, it just has the same sign as the sine function.
So many trig functions. Now i wanna make a website with dozens of examples on how to use all of them. Many caluclators support 12 trig functions (3 base functions * 2 for functional inverses * 2 for hyperbolics * 2 for multiplicative inverses). I can't believe there are more functions
When I first heard of these functions I was very surprised what kind of purpose leads to these definitions. I can´t remember all details but I´ve read that the haversine was important to calculate routes for ships. There a couple of (1-cos(theta))/2 terms, therefor they wrote hav(theta). Another reason for exterior secant or exterior cosecant and others was, that in special cases the taylor series in some intervall gets more beautiful. But for now I have to think for all the details. I leave that as an exercise for the reader. :P
Keep in mind that trig functions were very important in ship navigation and astronomy, long before calculators or Taylor series (which allow us to rather easily evaluate approximate values of trig functions by hand) were invented. If you were to need the value of, say, the tangent function at a specific angle you've determined to the accuracy of your measurement, you'd look it up in long reference works that just recorded known values - and these values were found through sum and half-angle formulae (since you could just break an angle like 34 degrees into 22.5 + 11.25 degrees, and after approximately evaluating the trig values of these angles - keep in mind they'd involve square roots, so you'd have had to approximate here as well, though it's not a huge source of error since there's an unreasonably efficient numerical method to evaluate square roots that was already known to the ancient Babylonians -, you could use addition formulae to find out the approximate value of the tangent for this angle). Most famously, a guy called Bürgi published an incredibly long and accurate table of sines - the _Canon Sinuum_ - in the 16th century that was unfortunately lost. Anyway, if all you have are lists of numerical values, it's actually pretty useful to define as many different trig functions as you can for all sorts of obscure purposes - from a modern perspective, you only really need the sine function, maybe cosine and tangent as well, but imagine having to evaluate something like the cosecant of 60 degrees per hand - you'd have to perform Euclidean division with a difficult number in the denominator. In this specific case, you could simplify the problem by rationalizing the denominator, but the trigonometric functions almost always return transcendental values, and so there's no way to find a technique that could even simplify the problem for most real numbers, let alone all of them.
This shouldn't be considered an alternative form of trigonometry (such as hyperbolic / spherical trigonometry) as these functions aren't ratios. If anything, they are nonlinear transformations from the basis of our normal trig functions.
This isn't considered alternative form. This is the same circular trigonometry. They're just unpopular. You don't need to be a ratio to be a trig function. You only need to relate the trangle to the circle. Also, negating and adding and subtracting 1 is quite linear.
11:55 Instead of using coordinates, you can also split the isosceles triangle with base crd(theta) in half, and then the end formula drops out much more easily.
@@leif1075 off the cuff, I'd have to say a piece of code for drawing measurements to a surface. Where he worked? It's huge, known worldwide. Because it wasn't a discussion on 3D graphics in 1978. Or was it? Or was it? Or was it?
for the cord one can also notice that the triangle whose base is the crdθ is Isosceles, i.e. it has two sides of length equal to 1. so we can cut it in half along its height, and from that it is already obvious that crdθ = 2sin(θ/2).
It's much easier to define crd(θ) if you drop a perpendicular bisector from circle center onto the chord. It divides the angle and chord in half. Now, in the right triangle formed by radius, half chord and perpendicular bisector, sin(θ/2) = 1/2 crd(θ). So that mean crd(θ) = 2sin(θ/2).
it's kind of cool that the equation for cord t = 2sin t/2 means you can bisect theta by drawing the chord, halving the chord by perpendicular bisector then drawing a line that joins the center of the circle with the point at the intersection of the bisector with the circle
Should there be an absolute value when we take the square root of sine squared? So crd(theta)=2|sin(theta/2)| ? Or is it useful to define it in such a way that it becomes negative for every other rotation around the circle?
Note that the range of the chord function is going to be from 0 to 2pi. then the range of the sine of half that value will, perforce, be nonnegative. While you're technically correct when suggesting the application of the absolute value function, it will be functionally moot. (If you wish to be sassy and define the range as -pi to +pi, recall that chord function measures a length, and so the abs() would be understood.)
If you just know the name you know nothing. I have been hearing this a lot lately, it's so true. I'm a CS grad and I don't recall these in any trig, calc, physics, linear, ... I was challenging myself working on real world problems and I had to re-learn relationships of geometry. Geometry is the KEY to mathematics. And in this journey I learned relationships of circles and triangle in circles, 4 sided objects in circles and more. I don't know where that "subject" matter is taught. I'd guess an advanced math degree where they actually show you what real math is, what geometry really is and not how to get to the next grade with fundamental nonsense. My profs always said, "don't memorize, derive the formula". It's so true. If you cannot derive, then you do not KNOW. And so I started applying that to many things. Such as. Such as PI. Okay, circumference over diameter. What is a line? It's only an idea and cannot be made real. Two lines of equal length. Impossible, can't be done, not even with a machine. You understand. Of course you do. So then, what is PI? You cannot measure the circumference. With a computer you can! No, PI is programmed as a constant number. With a string! Nope, you will always get a different measurement. How do you measure the diameter then? You cannot! So what is PI?! Irrational it is. So, PI is a number agreed upon. You cannot measure the circumference of a circle if you do not have PI already given as a constant. Therefor PI cannot be derived. So I ask, where did PI come from? That guy in a book? I really doubt it. And so I applied Sine and cosine, how to do it? I know : tan(angle) = Y/X. But that's a computer algorithm! Oh, the Taylor series! That's also a computer problem! And ancient ponderer of Philosophy did the Taylor series huh? And then a guy did tangent? Why is a slope not an angle? That's a real question. Why is not arctangent(PI/4) = 1? Why 360? Why 2*PI is a circle? Because circum/diameter is PI? I cannot fint anywhere on the net how to calculate SLOPE to RADIAN. Everything says see Table. The table was made by algorithm, right? And what is that so? We are NOT given the real math is the only logical and reasonable explanation. What say you? And would a PhD in math answer this question, and if so, why do I have to pay for a PhD to find out?
Pi can be defined as the zero of the sine Function, withch can be shown with a taylor series expansion. Or through the calculation of he area of a (semi-)sphere with the integral.
All random Trig Functions: 1-cosθ=Versed Sine (Written as versin/vers/ver) 0.5(verθ)=Half Versed Sine (Written as haversin/hav/hvs/even hv allegedly) 1-sinθ=Co-Versed Sine (Written as coversin/covers/cvs) 0.5(cvsθ)=Half Co-Versed Sine (Written as hacoversin/hacovers/hcv) 1+cosθ=Versed Cosine (Written as vercosin/vercos/vcs) 0.5(vcsθ)=Half Versed Cosine (Written as havercosin/havercos/hvc) 1+sinθ= Co-Versed Cosine (Written as covercosin/covercos/cvc) 0.5(cvcθ)=Half Co-Versed Cosine (Written as hacovercosin/hacovercos/hcc) secθ-1=Exterior Secant (Written as exsec) cscθ-1= Exterior Cosecant (Exterior Cosecant (Written as excsc) (In its simplest derivation): sqrt(sin^2θ+ver^2θ)= Chord (written as crd) Yes, all of the Haversine equivalents DO exist and are not made up (their existence is directly mentioned in Versine article in Wikipedia, under the section regarding inverse/arc functions) It's also good to know the historical explanation for the existence of Haversine functions was because of its particular usefulness in navigational calculation. More context on haversine can be seen through the Haversine Formula article on Wikipedia. so, in conclusion, stating the trigonometric function using their most commonly used and simplest abbreviation, we have: sin cos tan csc sec cot ver hvs cvs hcv vcs hvc cvc hcc exsec excsc crd This equals a total number of 17 trigonometric functions objectively, but is definitely up to each person on the validity of outdated or irrelevant functions.
Is the vertheta always measured from the point (1,0)? Like, say the angle is in the second quadrant, is it still measured from the right? Edit: Should’t the exsec be equal to sec - 1? I think you have a sign mistake...
it would be very easy to work out the last formula for crd(theta) by just using geometry. the cord is perpendicular to the radius of the circle at angle theta/2 and the radii at angles 0 and theta are equally far apart from the center of this cord. this means that the cord is twice of the distance from its center to the points where the other two radii cross the circle. and those are sin(theta/2), therefore crd(theta)=2sin(theta/2)
It seems to me that exsec theta goes to infinity as theta approaches pi/2. Your first operation with the ratio for similar triangles requires division by cos theta which approaches zero as theta approaches pi/2.
As much as I like these relations, I am really of the opinion that we already teach too many redundant trig functions. Reciprocal functions (sec, csc, cot) really don't need their own names and rules, and there is little value in memorizing their antiderivatives and such. There is already so much stuff we could pack into a math curriculum, there is no reason to jam it with 24 separate transcendental functions (including inverses) relating to circles and hyperbolas. But I'm glad that at least we no longer have the literally dozens of functions that existed in the past. If the only practical way to find a cosine is to look it up in a table, then having a table for 1 - cos x is valuable. But with calculators, that sort of thing is obsolete.
"You might think to yourself, 'Why haven't I heard of these trigonometric functions before?', and that's because they can all be easily described in terms of cosine and sine." ... so why do we learn about tangent, cotangent, secant, and cosecant? Those too can be easily described in terms of cosine and sine.
Because the applications, I guess. In general/basic schedules such as Calculus, we see reciprocal trig functions very often. The same can't be said about these ones, or at least they are not frequently related to the popular topics. It's a shame that they are somehow in the oblivion, by the way.
everything can be reduced to sin and cosine, correct but in calculus, there were a few that stood out that had to be retained ∫dx/(1+x^2) = arctan(x) + C which is elementary requiring tan(θ), further more accepting tan(θ) leads to requiring sec(θ). If anything it should be reduced to 4 only: sine, cosine, tangent, and secant. csc and cot only exist for merely completeness since secant gets the most attention anyways.
@@duckymomo7935 That's true, tangent and secant do come up more. I think it would be reasonable to stop there instead of also doing cotangent and cosecant. I do wonder if versine would come up more often in natural situations, if people still used it.
Justin Troyka Newton screwed up the notation for inverse functions by using the -1 exponent to mean “inverse” instead of “reciprocal”. Rather than saving millions of future students the confusion between inverses and reciprocals, the mathematical community compounded Newton's mistake by keeping his bad notation for inverse trig functions, which eventually necessitated the invention of names for the reciprocal trig functions once it was realised that they had practical applications. It's one of those situations-like pi versus tau-in which mathematicians chose to stick with established conventions that are objectively worse, simply because they don't believe the effort required to update and reinform everyone is worth it.
@@JonathonV Newton did fine there, actually, mathematicians just lazy and don't like to use parentheses. It's really easy to avoid any confusion by writing sin(x) instead of sin x, e.g. sin⁻¹(x) - inverse of a function, sin(x⁻¹) - function of a reciprocal, sin(x)⁻¹ - reciprocal of a function, and any combination can be easily obtained after that. Notation is fine, we should just make using brackets a norm and all confusion will be gone in no time.
Interesting. Here, these, as well as the csc, sec and cot functions don't seem to be part of any curriculum anymore, neither at highschool nor at university. And I studied maths (not as major, but still). I don't think they are in any way relevant today, as you have shown they can be easily built from the usual trigonometric functions. Still, it's interesting that they apparently served some purpose long ago
the point was more that you can get through higher mathematics, even trigonometry and geometry, without ever hearing of these. I mean you don't gain anything by using sec, cosec, etc. because the formulas are just as easy with just regular sin,cos,tan functions. Why should sec(x) be easier to understand than 1/cos(x) if you encounter it in a formula? Also,from a didactic point of view, it's also easier to teach and learn, because students are already confused by having three trigonomic functions, let alone 6 or 9 :-D But yeah, from a mathematical-historical point of view, it's definitely interesting to learn about them. But for everyday mathematics (even higher mathematics) they're not at all useful IMHO
According to your derivation dispayed in the board exsec(x)= 1-sec(x). But sec(x) is greater than or equal to one which follows that exec(x) is less than or equal to zero which is impossible. I think there is a mistake in simplifying. It should be exsec(x) = sec(x) - 1
Tangent is the line that connects the point (1,0) to the point made by 1 + exsec(). In other words the vertical leg of the larger triangle. The arc functions are just the inverse trigonometric functions. So arcsin() is the same as sin^(-1)(). Where plugging in an angle into sin() gives you a coordinate, plugging in the coordinate into arcsin() gives you the angle.
I'll try. First of all let's consider the equation cos(a)=tan(b) for a and b acute angles. We know that, b must be between 0 and 45, because tan(b), for b bigger than 45, is bigger than 1, and it can not be equal to cos(a). For the same line of reasoning, we know that a, b and c are in the set [0, 45]. Now, let's consider cos(a) and tan(a). We see that in this particular interval, cos(a) is bigger or equal than tan(a) (you can do that by simply looking at the graphs, or you can solve the inequality, that's up to you). Now we know that: cos(a)>/=tan(a). But we also know that cos(a)=tan(b), so we have: tan(b) >/=tan(a). Wich means that if cos(a)=tan(b), for b between 0 and 45, than b>/=a. Now let's go back to the orignial problem. We have: cos(a)=tan(b) (so a bigger or equal than b), cos(b)=tan(c) (so b bigger or equal than c), cos(c)=tan(a) (so c bigger or equal than a). Let's put all of this together, and we get: a
A more geometrical way to get crd(theta) is to observe that the other two sides of the triangle have length 1, so the triangle is isoc. Drop a line from the centre of the circle to the middle of the chord, and you get that half of the chord is sin(theta/2), and so the chord is 2sin(theta/2).
@@farfa2937 Not always, it was all done in terms of chords by the Greeks, it was the Indians who eventually made the sine the standard trigonometric function, though I believe they started out using the versine. In any case, the standard really is quite arbitrary as they can all be expressed in terms of each other.
if u said these f(x)s fell out of favour cuz they can be described in terms of sin n cos so why arent tan, cot etc also fallen out of favour? they can also be described in terms of sin n cos too
I suspect tan is still used because the inverse tan can (to my knowledge) not be written in terms of (inverse) sin or cos. As for cot, sec and cosec, the conflicting notation between powers of trig functions and inverse trig functions have probably caused people to use cot, sec and cosec instead. But in the Netherlands, cotangent, secant and cosecant are not learnt in high school maths.
. Micheal Penn . please subtitle to correct your error in exsec. Note sec>1, so your formula gives a negative length. Also, chord formula is much easier to derive, by bisecing the original triangle into two right triangles. chiord(x)= 2 sin(x/2) is self evident.
You lied, I did know of them. Well haver , vers and excec at least. For the spherical part, how were you doing a cos(a) when a was a length like 5 miles or whatever. Maybe I am trying to figure out the angle between here on Rochester when I know the distance from here to Toronto and Buffalo and themselves.
I was wondering about that too. It helps to think about spherical geometry. On a sphere, a "line" is a great circle (one whose center is the center of the sphere), since a great circle route (called a "geodesic") is the shortest distance between two points on a sphere. Since the sides of a spherical triangle must be "line segments", or geodesics, the sides are arcs of great circles. Now, think about what a geodesic is. It is an arc of a great circle, that is an arc on the sphere intercepted by a central angle (angle whose vertex is the center of the sphere). Since the sphere has radius 1, by definition of radian measure, the lengths of the arcs are just the radian measures of the central angles they are intercepted by. So cos(a) = cos(phi), where phi is the central angle (vertex at the center of the sphere), between the endpoints of the arc a. In this sense, the cosine of a side of a (spherical) triangle now makes sense. That is how I interpreted it, anyway.
@@mushroomsteve Yes, exactly. I had to think about it for a moment too, the "lengths" in spherical geometry are the angles that the segment of (great) circle it subtends at the centre of the sphere
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Btw, that vertical line parallel to the sine is the tangent. That's how we define tan θ, by extending the radius to intersect the tangent axis.
Yup, and this fact leads to a nice geometric proof that the limit sin(x)/x as x->0 is 1.
Yes indeed! Because you can approach sinx with x when close to zero (x = sinx = tanx), which is also geometrically evident as you said.
That's why it's named tangent then, it makes sense
@@suspendedsuplexchannel1000 they always have the same length so you can define it both ways. There's no "true" way to define things and this two ways to define tan can both be useful depending on the context. Where I came from we define tan exactly like that. It's actually very useful for rough mental estimation of tan value at any given angle, I would argue it's more useful for this then a standard definition in the West. In the end it doesn't matter, math is not about things themselves but about connections between those things. Or as Poincare sad it, "Mathematics is the art of giving the same name to different things".
It is, this is a great video BUT it is frustrating that he does not point out the tangent (and even willfully dances around it)
This is like trigonometric fanfiction
When you love trig so much you make up new functions to play with 😂
Trigonometric fanction 🫠
I've had a deep love for these ratios for more than 40 years. And I even befuddled my teachers with the more obtuse ones. Brilliant, this!
At 8:40, exsec = sec - 1. Just a minor typo, but I think I saw a similiar type of problem on math olympiad. Just cant remember the year and country
8:37 to 8:47 Oops, looks like you've flipped the order, should be exsec = sec - 1
Ah pheww, for a second I thought that only me notices. Yeah. He went to fast X)
Hahaha too*
Yeah he did
At 11:03 he also made anskitake right..you factor out a 2 not a 4..
@@leif1075 He put in a 1/2 to compensate.
sqrt(2-2*cos)
sqrt(2(1-cos))
sqrt(4*(1/2)*(1-cos))
2*sqrt((1/2)*(1-cos))
Haversine is my favorite little known trig function. Glad you brought it up Michael Penn. Very happy :)
is there also hyperbolic versions of these functions??
Yes. Google "hyperbolic versine"
The nice thing in math is: Just define it. :D
ah yes, inverse hyperbolic havercosine mmm
well done! first video in all of youtube that actually explains these...many thanks
If you are wondering what the cosine of a side of a spherical triangle means, referring to the spherical triangle he was drawing on the board: The sides of the triangle are arcs of great circles (geodesics) on the sphere. A great circle is a circle on the sphere, whose center is the center of the sphere. For example, if the Earth were exactly a sphere, all meridians of longitude would be great circles, and so would the equator.
This means the sides of the triangle are arcs intercepted by central angles of this unit sphere. Since the sphere is of radius 1, the radian measure of these central angles is precisely the lengths of the arcs they intercept. So that is why it makes sense to talk about the cosine of a "side", rather than an angle, of the spherical triangle.
Nice simplifications the unit circles make
mushroomsteve Thank you! 😀 I was just going to ask about that!
Transportation engineer here:
The first time I remember seeing exsec and versine was while studying sophomore geomatics. Until recently they were both still used when laying out horizontal roadway curves by transport surveyors (commonly called "staking the right-of-way/centerline"). Reason? You can't shoot "through" an arc, but you can through the chord(s). The distance from the chord to the midpoint of the arc is a multiple of the versine. Likewise, you may have only one means to reach the point where the highway tangents would meet (called the P.I.), so to get back to the curve, you need the exsec. The actual on-the-ground distances are called the middle ordinate and external distance respectively. Of course, the angle in question is equals (or is twice depending on reference) the angle of deflection (the direction you will be driving/riding minus the direction you previously were driving/riding), and we (the roadway engineers) provide this in our submittal packets to the surveyor and the contractor. The funny thing is that, with the advent of differential GPS, curves can be laid out much more precisely without worrying about where the chords, middle ordinates, and external distances are, so basically nobody even knows how to use exsec and versine anymore.
You cord function should have the absolute value of the sine(t/2)
the application of pythagoras' gives you the absolute value of the cord function, so |cord(t)|=2|sin t/2|. You can define it negative for negative angles, it just has the same sign as the sine function.
So many trig functions. Now i wanna make a website with dozens of examples on how to use all of them. Many caluclators support 12 trig functions (3 base functions * 2 for functional inverses * 2 for hyperbolics * 2 for multiplicative inverses). I can't believe there are more functions
When I first heard of these functions I was very surprised what kind of purpose leads to these definitions.
I can´t remember all details but I´ve read that the haversine was important to calculate routes for ships.
There a couple of (1-cos(theta))/2 terms, therefor they wrote hav(theta).
Another reason for exterior secant or exterior cosecant and others was, that in special cases the taylor series in some intervall gets more beautiful.
But for now I have to think for all the details. I leave that as an exercise for the reader. :P
Keep in mind that trig functions were very important in ship navigation and astronomy, long before calculators or Taylor series (which allow us to rather easily evaluate approximate values of trig functions by hand) were invented. If you were to need the value of, say, the tangent function at a specific angle you've determined to the accuracy of your measurement, you'd look it up in long reference works that just recorded known values - and these values were found through sum and half-angle formulae (since you could just break an angle like 34 degrees into 22.5 + 11.25 degrees, and after approximately evaluating the trig values of these angles - keep in mind they'd involve square roots, so you'd have had to approximate here as well, though it's not a huge source of error since there's an unreasonably efficient numerical method to evaluate square roots that was already known to the ancient Babylonians -, you could use addition formulae to find out the approximate value of the tangent for this angle). Most famously, a guy called Bürgi published an incredibly long and accurate table of sines - the _Canon Sinuum_ - in the 16th century that was unfortunately lost. Anyway, if all you have are lists of numerical values, it's actually pretty useful to define as many different trig functions as you can for all sorts of obscure purposes - from a modern perspective, you only really need the sine function, maybe cosine and tangent as well, but imagine having to evaluate something like the cosecant of 60 degrees per hand - you'd have to perform Euclidean division with a difficult number in the denominator. In this specific case, you could simplify the problem by rationalizing the denominator, but the trigonometric functions almost always return transcendental values, and so there's no way to find a technique that could even simplify the problem for most real numbers, let alone all of them.
@@beatoriche7301 _Nakanaideeee_
@@allaincumming6313 A fellow Umineko fan, I see ...
This shouldn't be considered an alternative form of trigonometry (such as hyperbolic / spherical trigonometry) as these functions aren't ratios. If anything, they are nonlinear transformations from the basis of our normal trig functions.
This isn't considered alternative form. This is the same circular trigonometry. They're just unpopular. You don't need to be a ratio to be a trig function. You only need to relate the trangle to the circle.
Also, negating and adding and subtracting 1 is quite linear.
They are actually ratios (length of a specific line to the radius of a circle).
best channel in youtube, thanks for all the content!!
11:55 Instead of using coordinates, you can also split the isosceles triangle with base crd(theta) in half, and then the end formula drops out much more easily.
Yeah, dropping a perpendicular on the cord is waaay more intuitive to me.
Yes, people, exsec(θ) = sec(θ) - 1.
Please post more about this.
8:45 that would work out to exsec(x) = sec(x) - 1, there was a factor of -1 introduced from thin air it seems
Hihi secx
Wow l gave this talk in 1978 to the McDonnell Douglas computer programmers .
How do these extra trig functions particularly help or are particularly relevant to computer programming, I'm curious?
@@leif1075 off the cuff, I'd have to say a piece of code for drawing measurements to a surface. Where he worked? It's huge, known worldwide. Because it wasn't a discussion on 3D graphics in 1978. Or was it? Or was it? Or was it?
for the cord one can also notice that the triangle whose base is the crdθ is Isosceles, i.e. it has two sides of length equal to 1. so we can cut it in half along its height, and from that it is already obvious that crdθ = 2sin(θ/2).
Versine is useful in elliptical geometry, such as for calculating distances on Earth, which in turn is useful for navigation or astronomy.
It's much easier to define crd(θ) if you drop a perpendicular bisector from circle center onto the chord. It divides the angle and chord in half. Now, in the right triangle formed by radius, half chord and perpendicular bisector, sin(θ/2) = 1/2 crd(θ). So that mean crd(θ) = 2sin(θ/2).
it's kind of cool that the equation for cord t = 2sin t/2 means you can bisect theta by drawing the chord, halving the chord by perpendicular bisector then drawing a line that joins the center of the circle with the point at the intersection of the bisector with the circle
i heard of these before, never used them though... do one on like really out there ones like covercosine lol
I seem to recall that Newton used a version of the chord function in the Principia.
Should there be an absolute value when we take the square root of sine squared? So crd(theta)=2|sin(theta/2)| ? Or is it useful to define it in such a way that it becomes negative for every other rotation around the circle?
Note that the range of the chord function is going to be from 0 to 2pi. then the range of the sine of half that value will, perforce, be nonnegative.
While you're technically correct when suggesting the application of the absolute value function, it will be functionally moot.
(If you wish to be sassy and define the range as -pi to +pi, recall that chord function measures a length, and so the abs() would be understood.)
Wow. Thanks. Have never heard of them before except for cord.
Sorry, but I think you made a sign error : exsec 𝜃 = 1 - sec 𝜃
If you just know the name you know nothing. I have been hearing this a lot lately, it's so true. I'm a CS grad and I don't recall these in any trig, calc, physics, linear, ... I was challenging myself working on real world problems and I had to re-learn relationships of geometry. Geometry is the KEY to mathematics. And in this journey I learned relationships of circles and triangle in circles, 4 sided objects in circles and more. I don't know where that "subject" matter is taught. I'd guess an advanced math degree where they actually show you what real math is, what geometry really is and not how to get to the next grade with fundamental nonsense. My profs always said, "don't memorize, derive the formula". It's so true. If you cannot derive, then you do not KNOW. And so I started applying that to many things. Such as. Such as PI. Okay, circumference over diameter. What is a line? It's only an idea and cannot be made real. Two lines of equal length. Impossible, can't be done, not even with a machine. You understand. Of course you do. So then, what is PI? You cannot measure the circumference. With a computer you can! No, PI is programmed as a constant number. With a string! Nope, you will always get a different measurement. How do you measure the diameter then? You cannot! So what is PI?! Irrational it is. So, PI is a number agreed upon. You cannot measure the circumference of a circle if you do not have PI already given as a constant. Therefor PI cannot be derived. So I ask, where did PI come from? That guy in a book? I really doubt it. And so I applied Sine and cosine, how to do it? I know : tan(angle) = Y/X. But that's a computer algorithm! Oh, the Taylor series! That's also a computer problem! And ancient ponderer of Philosophy did the Taylor series huh? And then a guy did tangent? Why is a slope not an angle? That's a real question. Why is not arctangent(PI/4) = 1? Why 360? Why 2*PI is a circle? Because circum/diameter is PI? I cannot fint anywhere on the net how to calculate SLOPE to RADIAN. Everything says see Table. The table was made by algorithm, right? And what is that so? We are NOT given the real math is the only logical and reasonable explanation. What say you? And would a PhD in math answer this question, and if so, why do I have to pay for a PhD to find out?
Pi can be defined as the zero of the sine Function, withch can be shown with a taylor series expansion. Or through the calculation of he area of a (semi-)sphere with the integral.
I have a problem suggestion: prove that when (a^2+b^2) / (ab-1) is an integer it'll always be the same integer. a,b are also integers.
I don't get what you mean by "is a constant". Because the value of (a^2+b^2)/(ab-1) clearly depends on the value of a and b
@@hach1koko My bad, I forgot something oops. When that expression results in an integer it'll always be the same integer.
8:40 isnt exsec(theta) = sec(theta) - 1, not 1 - sec(theta)?
Yes,it must be 1-sec(theta).
De acuerdo con ustedes. Saludos desde Argentina.
@@PabloPerez-er4og saludos de Irlanda también 👋
θ
All random Trig Functions:
1-cosθ=Versed Sine (Written as versin/vers/ver)
0.5(verθ)=Half Versed Sine (Written as haversin/hav/hvs/even hv allegedly)
1-sinθ=Co-Versed Sine (Written as coversin/covers/cvs)
0.5(cvsθ)=Half Co-Versed Sine (Written as hacoversin/hacovers/hcv)
1+cosθ=Versed Cosine (Written as vercosin/vercos/vcs)
0.5(vcsθ)=Half Versed Cosine (Written as havercosin/havercos/hvc)
1+sinθ= Co-Versed Cosine (Written as covercosin/covercos/cvc)
0.5(cvcθ)=Half Co-Versed Cosine (Written as hacovercosin/hacovercos/hcc)
secθ-1=Exterior Secant (Written as exsec)
cscθ-1= Exterior Cosecant (Exterior Cosecant (Written as excsc)
(In its simplest derivation): sqrt(sin^2θ+ver^2θ)= Chord (written as crd)
Yes, all of the Haversine equivalents DO exist and are not made up (their existence is directly mentioned in Versine article in Wikipedia, under the section regarding inverse/arc functions)
It's also good to know the historical explanation for the existence of Haversine functions was because of its particular usefulness in navigational calculation. More context on haversine can be seen through the Haversine Formula article on Wikipedia.
so, in conclusion, stating the trigonometric function using their most commonly used and simplest abbreviation, we have:
sin
cos
tan
csc
sec
cot
ver
hvs
cvs
hcv
vcs
hvc
cvc
hcc
exsec
excsc
crd
This equals a total number of 17 trigonometric functions objectively, but is definitely up to each person on the validity of outdated or irrelevant functions.
Is the vertheta always measured from the point (1,0)?
Like, say the angle is in the second quadrant, is it still measured from the right?
Edit:
Should’t the exsec be equal to sec - 1? I think you have a sign mistake...
it would be very easy to work out the last formula for crd(theta) by just using geometry. the cord is perpendicular to the radius of the circle at angle theta/2 and the radii at angles 0 and theta are equally far apart from the center of this cord. this means that the cord is twice of the distance from its center to the points where the other two radii cross the circle. and those are sin(theta/2), therefore crd(theta)=2sin(theta/2)
Very helpful explanation!
It seems to me that exsec theta goes to infinity as theta approaches pi/2. Your first operation with the ratio for similar triangles requires division by cos theta which approaches zero as theta approaches pi/2.
Nope, exsec(pi/2)=0.
Great video! Thank you!
Chords were extensively used before the advent of the other trig functions, especially in astronomy.
Might I ask why you have a photo of Field Marshall Erwin Rommel? It seems incongrous in a mathematical site.
@@roberttelarket4934 Why not?
I think I have seen versine (in terms of cosine) in the parametric equation of the cycloid.
Very good Sir
These are still around in engineering and science textbooks and monographs.
Thank god I'm not the only one who draws v's the same as r's
I wonder if haversine could be useful in making hoverboards
The exsec=sec-1 ... Thanks for this presentation.
Here's a fun challenge for some viewers which are doable with what you already have: find the inverse of as many of these as you can
As much as I like these relations, I am really of the opinion that we already teach too many redundant trig functions. Reciprocal functions (sec, csc, cot) really don't need their own names and rules, and there is little value in memorizing their antiderivatives and such. There is already so much stuff we could pack into a math curriculum, there is no reason to jam it with 24 separate transcendental functions (including inverses) relating to circles and hyperbolas. But I'm glad that at least we no longer have the literally dozens of functions that existed in the past.
If the only practical way to find a cosine is to look it up in a table, then having a table for 1 - cos x is valuable. But with calculators, that sort of thing is obsolete.
The chord actually looks quite useful
I really love your videos Mathematics is wonderfull❤.
"You might think to yourself, 'Why haven't I heard of these trigonometric functions before?', and that's because they can all be easily described in terms of cosine and sine."
... so why do we learn about tangent, cotangent, secant, and cosecant? Those too can be easily described in terms of cosine and sine.
Because the applications, I guess. In general/basic schedules such as Calculus, we see reciprocal trig functions very often. The same can't be said about these ones, or at least they are not frequently related to the popular topics. It's a shame that they are somehow in the oblivion, by the way.
everything can be reduced to sin and cosine, correct
but in calculus, there were a few that stood out that had to be retained
∫dx/(1+x^2) = arctan(x) + C which is elementary requiring tan(θ), further more accepting tan(θ) leads to requiring sec(θ). If anything it should be reduced to 4 only: sine, cosine, tangent, and secant.
csc and cot only exist for merely completeness since secant gets the most attention anyways.
@@duckymomo7935 That's true, tangent and secant do come up more. I think it would be reasonable to stop there instead of also doing cotangent and cosecant. I do wonder if versine would come up more often in natural situations, if people still used it.
Justin Troyka Newton screwed up the notation for inverse functions by using the -1 exponent to mean “inverse” instead of “reciprocal”.
Rather than saving millions of future students the confusion between inverses and reciprocals, the mathematical community compounded Newton's mistake by keeping his bad notation for inverse trig functions, which eventually necessitated the invention of names for the reciprocal trig functions once it was realised that they had practical applications.
It's one of those situations-like pi versus tau-in which mathematicians chose to stick with established conventions that are objectively worse, simply because they don't believe the effort required to update and reinform everyone is worth it.
@@JonathonV Newton did fine there, actually, mathematicians just lazy and don't like to use parentheses. It's really easy to avoid any confusion by writing sin(x) instead of sin x, e.g. sin⁻¹(x) - inverse of a function, sin(x⁻¹) - function of a reciprocal, sin(x)⁻¹ - reciprocal of a function, and any combination can be easily obtained after that. Notation is fine, we should just make using brackets a norm and all confusion will be gone in no time.
Very interesting, thank you for sharing!
Got it Today!
Love these facts!
So is the Limit as x tends to 0 of ver x /x = 0 , as 1-cos x =ver x?
Can anyone suggest any book that contains this archaic trigonometric functions and their derivatives?
Amazing math videos 🤩👏👏👏 Thanks for sharing your knowledge to us! New subscriber here 👍
Thank you.
exsec n=|sec n|-1,ver n=1-|cos n|,hav n=(ver n)/2,crd n=|n| mod tau+|sin n|
Interesting. Here, these, as well as the csc, sec and cot functions don't seem to be part of any curriculum anymore, neither at highschool nor at university. And I studied maths (not as major, but still). I don't think they are in any way relevant today, as you have shown they can be easily built from the usual trigonometric functions. Still, it's interesting that they apparently served some purpose long ago
Although I don't know if that's the case for other countries. At least here in Brazil, we still study those while in highschool.
In British Columbia too.
the point was more that you can get through higher mathematics, even trigonometry and geometry, without ever hearing of these. I mean you don't gain anything by using sec, cosec, etc. because the formulas are just as easy with just regular sin,cos,tan functions. Why should sec(x) be easier to understand than 1/cos(x) if you encounter it in a formula? Also,from a didactic point of view, it's also easier to teach and learn, because students are already confused by having three trigonomic functions, let alone 6 or 9 :-D
But yeah, from a mathematical-historical point of view, it's definitely interesting to learn about them. But for everyday mathematics (even higher mathematics) they're not at all useful IMHO
Is this a topic of college or university???
Plzzz reply...I really love mathematics
So interesting...keep it up
In 8:45 it should be sec(theta)-1.You have written it oppsite.
According to your derivation dispayed in the board exsec(x)= 1-sec(x). But sec(x) is greater than or equal to one which follows that exec(x) is less than or equal to zero which is impossible. I think there is a mistake in simplifying. It should be exsec(x) = sec(x) - 1
Love your videos!
What about the tan and all the arc versions?
Tangent is the line that connects the point (1,0) to the point made by 1 + exsec(). In other words the vertical leg of the larger triangle.
The arc functions are just the inverse trigonometric functions. So arcsin() is the same as sin^(-1)(). Where plugging in an angle into sin() gives you a coordinate, plugging in the coordinate into arcsin() gives you the angle.
i didn't know it had a name but i always thought the 'versine' should have one, because of the compton shift formula
So why have the exsec, etc. gone out of use?
Is this how I can cut hours out of my IRL Any% SpeedRun
I've heard of versin but not any of the others
Glad to know that
It is interesting
Look for trig app in in Android or apple store
If a,b,c are acute angles such that cosa=tanb , cosb=tanc , cosc=tana then prove that sina=sinb=sinc=(root5-1)/2
Close ish from golden ratio
Blackdeath39 Muffin its -1/phi
I'll try. First of all let's consider the equation cos(a)=tan(b) for a and b acute angles. We know that, b must be between 0 and 45, because tan(b), for b bigger than 45, is bigger than 1, and it can not be equal to cos(a). For the same line of reasoning, we know that a, b and c are in the set [0, 45]. Now, let's consider cos(a) and tan(a). We see that in this particular interval, cos(a) is bigger or equal than tan(a) (you can do that by simply looking at the graphs, or you can solve the inequality, that's up to you). Now we know that: cos(a)>/=tan(a). But we also know that cos(a)=tan(b), so we have: tan(b) >/=tan(a). Wich means that if cos(a)=tan(b), for b between 0 and 45, than b>/=a. Now let's go back to the orignial problem. We have: cos(a)=tan(b) (so a bigger or equal than b), cos(b)=tan(c) (so b bigger or equal than c), cos(c)=tan(a) (so c bigger or equal than a). Let's put all of this together, and we get: a
what are these other functions that you speak of ? Also define these functions ?
Good
A more geometrical way to get crd(theta) is to observe that the other two sides of the triangle have length 1, so the triangle is isoc. Drop a line from the centre of the circle to the middle of the chord, and you get that half of the chord is sin(theta/2), and so the chord is 2sin(theta/2).
As already noted in other comments there is a mistake with the negative 1 at 8:37
A beautiful isosceles triangle completely ignored.
Nice
cos(t) = sin(pi/2 - t), but most of us don't go that far when it comes to reducing the number of trig functions.
- Wait, is it all sin?
- Always has been...
@@farfa2937 Not always, it was all done in terms of chords by the Greeks, it was the Indians who eventually made the sine the standard trigonometric function, though I believe they started out using the versine. In any case, the standard really is quite arbitrary as they can all be expressed in terms of each other.
@@costakeith9048 That'd make for a terrible meme line tho.
These videos make me feel really dumb... but then again I just completed high school last month so I think it’s ok that I don’t know this stuff
if u said these f(x)s fell out of favour cuz they can be described in terms of sin n cos so why arent tan, cot etc also fallen out of favour? they can also be described in terms of sin n cos too
I suspect tan is still used because the inverse tan can (to my knowledge) not be written in terms of (inverse) sin or cos. As for cot, sec and cosec, the conflicting notation between powers of trig functions and inverse trig functions have probably caused people to use cot, sec and cosec instead.
But in the Netherlands, cotangent, secant and cosecant are not learnt in high school maths.
. Micheal Penn . please subtitle to correct your error in exsec. Note sec>1, so your formula gives a negative length. Also, chord formula is much easier to derive, by bisecing the original triangle into two right triangles.
chiord(x)= 2 sin(x/2) is self evident.
wait, they have functions for triangles?
Absolute value? 11:28
I already knew some of them
cool! imma gonna integrate them!
...
oh. that’s just... not intuitive.
there are more .... exterior lines ❤️
What else have they hidden from us?
Did you know Sir Isaac Newton had testicles?
But wait there’s more!
These seem to be more trouble than they're worth.
They released DLC to trig
I already study it myself, I calcul the function in function of the sin and cos lmao
Exsec(x) = sec(x) - 1
You lied, I did know of them.
Well haver , vers and excec at least.
For the spherical part, how were you doing a cos(a) when a was a length like 5 miles or whatever.
Maybe I am trying to figure out the angle between here on Rochester when I know the distance from here to Toronto and Buffalo and themselves.
I was wondering about that too. It helps to think about spherical geometry. On a sphere, a "line" is a great circle (one whose center is the center of the sphere), since a great circle route (called a "geodesic") is the shortest distance between two points on a sphere. Since the sides of a spherical triangle must be "line segments", or geodesics, the sides are arcs of great circles.
Now, think about what a geodesic is. It is an arc of a great circle, that is an arc on the sphere intercepted by a central angle (angle whose vertex is the center of the sphere). Since the sphere has radius 1, by definition of radian measure, the lengths of the arcs are just the radian measures of the central angles they are intercepted by. So cos(a) = cos(phi), where phi is the central angle (vertex at the center of the sphere), between the endpoints of the arc a. In this sense, the cosine of a side of a (spherical) triangle now makes sense. That is how I interpreted it, anyway.
@@mushroomsteve Yes, exactly. I had to think about it for a moment too, the "lengths" in spherical geometry are the angles that the segment of (great) circle it subtends at the centre of the sphere
Worst sine or versine😂😂
Ironically, or interestingly, sine actually already means "chord". So the "chord" function in this video is a confusion.
but he didnt define the angles a, b, c... only the sides a, b, c.
can you prove this:-
1-1/5+1/7-1/11+1/13-.......infinity=pi/2root(3)
you made a mistake in video : exsec theta = sec theta - 1 ; not 1 - sec theta as you indicated in video
11:56
remarkable consistency