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[sup]-1/sup 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.
Why not as long as everyone agrees on it?
Because it’s easier.
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.
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.
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)][sup]2[/sup] is so unwieldy. Anything that can clear up sin[sup]-1[/sup]x being arcsin or cosecant is good.
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.
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.
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?
To make your post look like this:
sin[sup]-1/sup
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.
I remember that in my first Calculus exam of the year my teacher wrote on my paper that sin[sup]-1[/sup] and arcsin have different domains. I had written sin[sup]-1[/sup] for one of my answers when she thought arcsin was appropriate. I didn’t lose points for that, though.
Splanky, I think, perhaps, your teacher meant that sin[sup]-1/sup 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.
There are lots of equations in math and physics where the sin[sup]2[/sup] (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 sin[sup]2[/sup] (x) you wrote out (sin (x))[sup]2[/sup].
There is no difference between sin[sup]-1[/sup] 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.
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.
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.
To the OP: you’re right that the “exponents” in sin[sup]2[/sup] and sin[sup]-1[/sup] mean different things, but this is one place where consistency of notation is sacrificed to convenience. If you want to write [sin (x)][sup]2[/sup] instead of sin[sup]2[/sup]x, you go right ahead, but that gets to be a real pain when the sin[sup]2[/sup]'s fly thick and fast as they sometimes do.
Because of this potential for confusion, some books/teachers avoid using the sin[sup]-1[/sup] notation altogether, using arcsin instead. But in math, the [sup]-1[/sup] 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[sup]-1/sup is commonly understood to denote the inverse function of that function, not 1/f(x); so the sin[sup]-1[/sup] notation is consistent with this.
Partly the sin[sup]2/sup notation comes from the way it’s stated. “sine-ex-squared” could easily be confused for sin(x[sup]2[/sup]), so people say “sine-squared-ex”, which lends itself to writing as sin[sup]2/sup.
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 = g[sub]1/sub
y = g[sub]2/sub
z = g[sub]3/sub
f[sub]u/sub = f[sub]x/subg[sub]1u/sub + f[sub]y/subg[sub]2u/sub + f[sub]z/subg[sub]3u/sub
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 = Sum[sub]k=1[/sub][sup]n[/sup] Df/Dx[sub]k[/sub] Dx[sub]k[/sub]/Du
Or maybe you haven’t see the relation to the total differential. Given any manifold and a coordinate patch with coordinates {x[sup]k[/sup]}[sub]1<=k<=n[/sub] the fiber of the cotangent bundle above a point has a canonical basis {dx[sup]k[/sup]}[sub]1<=k<=n[/sub]. Given a function f on the manifold, its total differential at the point is
df(p) = Sum[sub]k=1[/sub][sup]n[/sup] Df/Dx[sub]k[/sub]|[sub]p[/sub] dx[sup]k[/sup]
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).
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.
Good point, Mathochist! I hadn’t thought of it, but you may have mentioned the best reason of all: sin[sup]2[/sup] x is easier to pronounce in an unambiguous way than [sin (x)][sup]2[/sup].
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.