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View Full Version : Why is [sin(x)]² written as sin²(x)?


TJdude825
05-29-2004, 01:23 AM
In my algebra class, we've been using sin²(x) to mean the square of sin(x), that is, [sin(x)]². By this logic, sin-1(x) should equal 1/sin(x) or csc(x), which it certainly does not. I'm not sure what I feel sin²(x) should mean. Maybe sin(sin(x))*, but the way we are writing it seems very strange to me. What's up with that?

*I realize that this is silly because sin(x) is a ratio, not an angle, and therefore it is meaningless to take the sine again.

David Simmons
05-29-2004, 01:24 AM
Why not as long as everyone agrees on it?

Smeghead
05-29-2004, 01:45 AM
Because it's easier.

ultrafilter
05-29-2004, 01:57 AM
*I realize that this is silly because sin(x) is a ratio, not an angle, and therefore it is meaningless to take the sine again.

No one outside of high school thinks of the sine as having to do anything with triangles. sin(sin(x)) is perfectly well-defined, if not something you encounter regularly.

The notation in question probably took off because it's quite common to square trigonometric functions, and very rare to compose them.

wolf_meister
05-29-2004, 02:03 AM
I always thought it was written sin²(x) as opposed to sin(x)² to avoid the confusion in thinking that you are taking the sine after you square the angle.
For example, sin(5)² could be thought of as sin(25).
By writing sin²(5) it eliminates this ambiguity.

Shade
05-29-2004, 03:54 AM
Yeah, it's arbitrary AFAIK. Can you suggest a better way? Seriously, if you can think of a less confusing and more convenient notation I'll use it. But [sin(x)]2 is so unwieldy. Anything that can clear up sin-1x being arcsin or cosecant is good.

Napier
05-29-2004, 07:31 AM
I agree completely with the OP that this is confusing and don't see anything wrong with [sin(x)]2. Squaring the trig functions already implies a level of effort that two extra parenthesis or bracket characters aren't about to exceed. The idea that (sin^n(x)) = (sin(x))^n when n ne -1 and arcsin(x) when n=1 is just plain bad notation.

The calculus notation d(y)/d(x) and especially d^2(y)/d(x)^2 is also just plain bad.

I used Mathematica and Maple for a few years and I liked the internal consistency of the notation systems there.

BobLibDem
05-29-2004, 07:38 AM
I've always hated the notation sin -1 (x). (Sorry I don't know how to do superscripts). To me, this should be the same as 1/sin(x). I like the syntax Arcsin(x) much better as it seems more direct and unambiguous.

aerodave
05-29-2004, 08:06 AM
The calculus notation d(y)/d(x) and especially d^2(y)/d(x)^2 is also just plain bad.

I don't see the problem with the derivative notations you posted. They describe perfectly well what's going on.

Could you possibly suggest a better alternative?

Musicat
05-29-2004, 09:03 AM
I've always hated the notation sin -1 (x). (Sorry I don't know how to do superscripts). To make your post look like this:

sin-1(x)

Write your post like this, but replace the curly braces {} with straight ones []:

sin{sup}-1{/sup}(x)

The "sup" code pair surrounds the text you want to be superscripted. "Sub" works the same way for subscripting.

Splanky
05-29-2004, 09:08 AM
I remember that in my first Calculus exam of the year my teacher wrote on my paper that sin-1 and arcsin have different domains. I had written sin-1 for one of my answers when she thought arcsin was appropriate. I didn't lose points for that, though.

iwakura43
05-29-2004, 10:23 AM
Splanky, I think, perhaps, your teacher meant that sin-1(x) has a different range than Arcsin(x) (note the capital A). Arcsin and Arccos have ranges 2π wide (Arctan is π wide), while the domain for the lower-case ones (which I assume to be equivalent to the -1 notation functions) have infinite ranges.

I'm only a high school math student, though, so feel free to correct me.

Dr. Lao
05-29-2004, 11:11 AM
There are lots of equations in math and physics where the sin2 (x) function appears. Often it appears multiple times in a single equation. It would be a pain and would clutter up already pretty cluttered equations if everytime you wanted to show sin2 (x) you wrote out (sin (x))2.

emarkp
05-29-2004, 11:30 AM
Splanky, I think, perhaps, your teacher meant that sin-1(x) has a different range than Arcsin(x) (note the capital A). Arcsin and Arccos have ranges 2π wide (Arctan is π wide), while the domain for the lower-case ones (which I assume to be equivalent to the -1 notation functions) have infinite ranges.There is no difference between sin-1 and Arcsin. They are the same function. And there's no difference between Arcsin and arcsin.

Since the range of sin/cos is -1 to 1, the domain of arcsin/arccos is also -1 to 1.

Orbifold
05-29-2004, 01:39 PM
I always thought it was written sin²(x) as opposed to sin(x)² to avoid the confusion in thinking that you are taking the sine after you square the angle.
For example, sin(5)² could be thought of as sin(25).
By writing sin²(5) it eliminates this ambiguity.

It's not uncommon to write sin(x) without the parentheses: for example, to write "sin pi/6" instead of "sin(pi/6)". Which does, I think, make wolfmeister's theory plausible. Of course, that's just using one unfortunate notational convention to justify another, but there you go.

Thudlow Boink
05-29-2004, 02:10 PM
No one outside of high school thinks of the sine as having to do anything with triangles.I don't think this is a fair comment, because (1) the OP didn't say anything about triangles, just ratios, and (2) once you get past high school math, sine is no longer defined in terms of triangles nor does it refer only to triangles, but the application to right triangles is hardly obsolete.

Thudlow Boink
05-29-2004, 02:20 PM
To the OP: you're right that the "exponents" in sin2 and sin-1 mean different things, but this is one place where consistency of notation is sacrificed to convenience. If you want to write [sin (x)]2 instead of sin2x, you go right ahead, but that gets to be a real pain when the sin2's fly thick and fast as they sometimes do.

Because of this potential for confusion, some books/teachers avoid using the sin-1 notation altogether, using arcsin instead. But in math, the -1 is used to denote an inverse, not just to mean "to the negative one power"—which is, after all, a kind of inverse: the multiplicative inverse of a number. But if f(x) is some function, f-1(x) is commonly understood to denote the inverse function of that function, not 1/f(x); so the sin-1 notation is consistent with this.

Mathochist
05-29-2004, 02:33 PM
I agree completely with the OP that this is confusing and don't see anything wrong with [sin(x)]2. Squaring the trig functions already implies a level of effort that two extra parenthesis or bracket characters aren't about to exceed. The idea that (sin^n(x)) = (sin(x))^n when n ne -1 and arcsin(x) when n=1 is just plain bad notation.

The calculus notation d(y)/d(x) and especially d^2(y)/d(x)^2 is also just plain bad.

I used Mathematica and Maple for a few years and I liked the internal consistency of the notation systems there.

Partly the sin2(x) notation comes from the way it's stated. "sine-ex-squared" could easily be confused for sin(x2), so people say "sine-squared-ex", which lends itself to writing as sin2(x).

Now, as to your silly assertion that the differential notation is "bad", have you ever tried to write out the chain rule in multivariable calculus without it?

f(x,y,z)
x = g1(u,v,w)
y = g2(u,v,w)
z = g3(u,v,w)
fu(u,v,w) = fx(g1(u,v,w),g2(u,v,w),g3(u,v,w))g1u(u,v,w) + fy(g1(u,v,w),g2(u,v,w),g3(u,v,w))g2u(u,v,w) + fz(g1(u,v,w),g2(u,v,w),g3(u,v,w))g3u(u,v,w)

As opposed to (using D instead of \partial, since we have no TeX interpreting here)

Df/Du = Df/Dx Dx/Du + Df/Dy Dy/Du + Df/Dz Dz/Du

Or if f depends on n variables, each of which depends on u

Df/Du = Sumk=1n Df/Dxk Dxk/Du

Or maybe you haven't see the relation to the total differential. Given any manifold and a coordinate patch with coordinates {xk}1<=k<=n the fiber of the cotangent bundle above a point has a canonical basis {dxk}1<=k<=n. Given a function f on the manifold, its total differential at the point is

df(p) = Sumk=1n Df/Dxk|p dxk

Or how about operator notation? The maps sending a differentiable function of n variables to its directional derivatives generate an algebraic structure which can be studied independantly of the functions themselves. Clearly it pays to have a notation for the operator without an argument just as it pays to have a notation for the function f independantly of its evaluations f(x).

Mathochist
05-29-2004, 02:48 PM
There is no difference between sin-1 and Arcsin. They are the same function. And there's no difference between Arcsin and arcsin.

Since the range of sin/cos is -1 to 1, the domain of arcsin/arccos is also -1 to 1.

Oh, next you'll tell us that Log and log are the same.

ultrafilter pointed this out earlier: trigonometry in the "real world" has nothing to do with triangles. When you drop in complex arguments you'll understand why your range argument fails.

Anyhow, traditionally Arcsin is used to denote the principal value of the inverse of sine. The sine sends any two numbers differing by 2? to the same value, so it cannot be properly inverted. A choice must be made, and saying Arcsin specifies the choice.

arcsin is technically only defined on the Riemann surface of the sine function, which is a branched cover of C. There is a "principal" branch which looks like C itself, and Arcsin is the restriction of arcsin to this branch. The casualty is that one must make "branch cuts" along which the restricted function fails even to be continuous, let alone holomorphic. Traditionally these are taken for Arcsin to be rays along the real line extending positively from 1 and negatively from -1.

Now, the catch is that not every book makes this distiction, or makes it in the same order. Still, every decent author will make clear when he is referring to the inverse sine or to its principal value.

Thudlow Boink
05-29-2004, 02:50 PM
Good point, Mathochist! I hadn't thought of it, but you may have mentioned the best reason of all: sin2 x is easier to pronounce in an unambiguous way than [sin (x)]2.

Napier, what's wrong with dy/dx? What would you rather use? There are several different notations for the derivative (see Mathochist's post), and which one is best depends on the context.

TJdude825
05-29-2004, 03:33 PM
Thanks for all the replies. I guess strictly logical and consistent conventions are not as important as making sense and not using up tons of paper writing ()'s. In fact, now that I think about it, I've been writing (3 + ½) as (3½) for years, and nobody would see that and try to multiply it out to 1.5.
Another little bit of ignorance fought and conquered!

Hyperelastic
05-29-2004, 03:44 PM
Napier, what's wrong with dy/dx? What would you rather use? There are several different notations for the derivative (see Mathochist's post), and which one is best depends on the context.

In continuum mechanics, we use u,x to denote the partial derivative of u with respect to x.

d2u/dydx is then u,yx

It's much more compact, unambiguous, and does not perpetuate the erroneous notion that the derivative is a ratio.

Mathochist
05-29-2004, 03:58 PM
Thanks for all the replies. I guess strictly logical and consistent conventions are not as important as making sense and not using up tons of paper writing ()'s. In fact, now that I think about it, I've been writing (3 + ½) as (3½) for years, and nobody would see that and try to multiply it out to 1.5.
Another little bit of ignorance fought and conquered!

Incidentally, there have been other notational conventions that arose from scarcity. The famous "Einstein summation convention" came about because in Einstein's papers on relativity, every other formula would involve at least one (sometimes two, three, or more) summation and the typesetters very quickly ran out of sigmas. The convention arose that whenever the same index showed up twice, it would be summed over the appropriate range. Often it was restricted to an upper and lower pair of indices, reflecting that one would be indexing coordinates on a tangent space and the other on a cotangent space. Real mathematicians now work without coordinates as much as possible, but you'll still find physicists and engineers index-juggling away.

As an example, the Riemann curvature tensor is (as far as GR is concerned) a rank four tensor, which has four indices.

Rijkl

Often it's written with the first index raised

Rijkl = gimRmjkl

Where g is the "metric tensor" and the summation convention means that the right hand side should be evaluated for m = 0, 1, 2, and 3 and all four of these values added together. Then one can define the Ricci curvature as

Rjl = Rijil

Then raise the first index and contract again to get the Ricci scalar curvature

R = Rll = gljRjl

So already I've saved five summation symbols by using the convention (see if you can see where they should go).

Mathochist
05-29-2004, 04:04 PM
In continuum mechanics, we use u,x to denote the partial derivative of u with respect to x.

d2u/dydx is then u,yx

It's much more compact, unambiguous, and does not perpetuate the erroneous notion that the derivative is a ratio.

Actually, you've got it backwards. Again using D for \partial:

D2u/DyDx = D/Dy D/Dx u = D/Dy u,x = u,xy

It's only in flat manifolds that mixed partials commute.

Anyhow, I'd be interested to see what notion of derivative you use in continuum mechanics that isn't defined by a limit of a ratio. Yes, there's a functorial definition, but that's so abstract it's almost meaningless to calculate with in applications.

tim314
05-29-2004, 07:53 PM
Thanks for all the replies. I guess strictly logical and consistent conventions are not as important as making sense and not using up tons of paper writing ()'s. In fact, now that I think about it, I've been writing (3 + ½) as (3½) for years, and nobody would see that and try to multiply it out to 1.5.
Another little bit of ignorance fought and conquered!
I for one am so used to writing 3 + ½ as 7/2 that I am always tempted to interpret 3½ as 31/2. Especially because some people actually type three halves as 31/2, without bothering to include a space after the three.

Can anyone tell me why high schoolers are taught that a fraction whose numerator exceeds its denominator is "improper"? Is there any technical field where this convention is actually followed?

Mathochist
05-29-2004, 08:06 PM
Can anyone tell me why high schoolers are taught that a fraction whose numerator exceeds its denominator is "improper"? Is there any technical field where this convention is actually followed?

High schoolers? If they haven't seen fractions by high school, something's dreadfully wrong. Anyhow, the immediate benefit is seeing where a given fraction falls with respect to the integers. 3678943578956/23718567423 doesn't tell as much at a glance as 155 2565628391/23718567423. Admittedly though, this is far outweighed in practice by the algorithmic benefits of "improper" fractions and the name "improper" is rather a loaded one.

Shade
05-29-2004, 08:22 PM
High schoolers? If they haven't seen fractions by high school, something's dreadfully wrong. Anyhow, the immediate benefit is seeing where a given fraction falls with respect to the integers. 3678943578956/23718567423 doesn't tell as much at a glance as 155 2565628391/23718567423. Admittedly though, this is far outweighed in practice by the algorithmic benefits of "improper" fractions and the name "improper" is rather a loaded one.But to a mathematician 'proper' isn't really loaded - it basically means 'non-pathological' or 'a normal sort of one', yes? Of course, that doesn't really explain it since mathematicians never and school children do use 'improper' and anyhow, improper fractions aren't pathological in the usual sense, but that could have been in the mind of whoever coined the term, couldn't it?

kushiel
05-29-2004, 10:30 PM
This just aggravates me when I get a question on, say, derivatives and it is written sinx2 and when I get confused, the teacher tells me to look at it as sin2x. Why not just do it that way in the first place? I mean, everyone sees it differently, but why isn't there a standard?

Mathochist
05-29-2004, 11:10 PM
But to a mathematician 'proper' isn't really loaded - it basically means 'non-pathological' or 'a normal sort of one', yes? Of course, that doesn't really explain it since mathematicians never and school children do use 'improper' and anyhow, improper fractions aren't pathological in the usual sense, but that could have been in the mind of whoever coined the term, couldn't it?

Well, it tends more to be used as "nontrivial". A proper subset is a subset other than the whole set itself. A proper ideal of a ring is an ideal which is neither the zero ideal or the entire ring. I'd be willing to bet that "improper fraction" has always been used in teaching basic arithmetic and never in "real mathematics".

It may seem pretty silly to have the people setting the basic math curricula and the people actually doing mathematics be separate groups, but remember that the last time mathematicians were allowed to decide how arithmetic should be taught we got "new math": so simple that only a child can do it.

Hyperelastic
05-29-2004, 11:45 PM
Actually, you've got it backwards. Again using D for \partial:

D2u/DyDx = D/Dy D/Dx u = D/Dy u,x = u,xy

It's only in flat manifolds that mixed partials commute.

Anyhow, I'd be interested to see what notion of derivative you use in continuum mechanics that isn't defined by a limit of a ratio. Yes, there's a functorial definition, but that's so abstract it's almost meaningless to calculate with in applications.

OK, if you want to get picky, in classical continuum mechanics a body is a three-dimensional differentiable manifold which must be viewed in its configurations. A configuration is a smooth homeomorphism of the body onto a region of plain old Euclidean 3-space. So we never have to deal with anything but a flat manifold. That is why we can use the subscript notation without getting confused, and we can always assume that mixed higher partials commute. Researchers in relativistic continuum mechanics, of whom there are a very, very few, use a different notation.

Also, a limit of a ratio is not the same thing as a ratio.

TJdude825
05-30-2004, 02:17 AM
And the thread went over my head (again) at Mathochist's last post, but I would like to say something about the word "improper." The way I understood it, it was just a name, the same way we call numbers irrational, complex, transcendental, etc. We're not saying that 3+2i is really a "made up" number, we just call it imaginary because it's useful to call it that. In the same way, I never got the impression that improper fractions were bad or undesirable. We just called 3/2 improper, and 1 1/2 mixed. I think every teacher I've had would accept either form as a correct answer, unless they specifically said to write one or the other. So I suppose I'd agree that "improper" is a "loaded" name, but so are several other names that mathematicians actually do use outside of high school.

tim314
05-30-2004, 12:32 PM
High schoolers? If they haven't seen fractions by high school, something's dreadfully wrong. Anyhow, the immediate benefit is seeing where a given fraction falls with respect to the integers. 3678943578956/23718567423 doesn't tell as much at a glance as 155 2565628391/23718567423. Admittedly though, this is far outweighed in practice by the algorithmic benefits of "improper" fractions and the name "improper" is rather a loaded one.

What I meant was not that they learn the term "improper fraction" in high school. (Even though, looking back at my comment, that is how I said it.) What I mean is why are they *still* using mixed numbers in high school. I see what you're saying about seeing where fractions lie relative to the integers, but typically in high school math classes students are allowed to use calculators (if I remember correctly), so they can just type the fraction in and instantly see it in decimal form. I'm just a bit bothered by the fact that students get to college and are still worrying about things like converting improper fractions to mixed numbers or rationalizing their denominators, rather than just putting the answers in the simplest looking form.

I think by high school the students should be learning to write mathematical expressions the way mathematicians, physicists, etc. would actually write them.

Mathochist
05-30-2004, 02:35 PM
What I meant was not that they learn the term "improper fraction" in high school. (Even though, looking back at my comment, that is how I said it.) What I mean is why are they *still* using mixed numbers in high school.

I'm just a bit bothered by the fact that students get to college and are still worrying about things like converting improper fractions to mixed numbers or rationalizing their denominators, rather than just putting the answers in the simplest looking form.

This sounds nothing like any math class beyond "pre-algebra" that I've seen. No algebra teacher I've ever talked with has insisted on one form or another. Further, I've been teaching college freshmen for quite a few years now and with the exception of some heavily remedial students I tutored as an undergraduate, none of them have "worried" at all about such things. Where are you getting this impression of yours?

David Simmons
05-30-2004, 02:41 PM
I think some people don't like the form 124/51 just as it was considered bad form when I was at the University of Iowa to leave a radical in the denominator.

Mathochist
05-30-2004, 02:48 PM
I think some people don't like the form 124/51 just as it was considered bad form when I was at the University of Iowa to leave a radical in the denominator.

Now I must ask which course this was and when it was taught. Both could be very enlightening.

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