4 ms·

The sin^n x notation is bad. I don't blame the undergrads there. So we learn the following fact: sin^{-1} x = y => x = sin

The sin^n x notation is bad. I don't blame the undergrads there. So we learn the following fact: sin^{-1} x = y => x = sin y Being enthusiastic new algebra students, we presume know this must work by applying sine to both sides: sin^{1} sin^{-1} x = sin^{1} y => sin^{0} x = sin y here sin^{0} is zero applications of sine to x => x = sin y Noting we could start at line 2, and apply sin^{-1} to both side also, we have now learnt that: sin^{a} sin^{b} x = sin^{a+b} x if a = +/- 1 and b = -/+ 1. Presumably, if notation is at all sane, the rule applies to other values of a and b so: sin^2 x = sin sin x (? Surely!) Right? No. sin^2 x = (sin x)^2 and sin sin x = (has no other name) No wonder students get confused. The notation is trying it's darnedest to confuse them.

Tainnor 6y ago
Yes, this is definitely a problem. I would just do away with the sin^{-1} notation (as, it seems, many textbooks already do) since we have the perfectly acceptable alternative "arcsin". It's also not a very good notation since it's trying to imply that the sin function has an inverse, but it doesn't. That's why "sin^{-1}(sin(x)) = x" is not even right in general. The inverse only exists on specific subintervals, and it's also off by a multiple of pi, depending on that subinterval. "arcsin" is then defined as the inverse of sin, restricted to the interval [-pi/2,pi/2]. Of course, the bigger issue here is that f^n for any function f is inherently ambiguous, because it could refer either to the (pointwise) multiplication operation or to the composition operation.
- Tomminn 6y ago
  This is a very good point. The sine function obviously doesn't have a technical inverse on any interval where it has two values. The notation does make me forget this sometimes. Of course, the situation is different from many other functions without inverses, because the set of all valid inversions can be trivially generated from one solution. Just put a mirror at pi/2 and -pi/2.

Tainnor 6y ago

Yes, this is definitely a problem. I would just do away with the sin^{-1} notation (as, it seems, many textbooks already do) since we have the perfectly acceptable alternative "arcsin". It's also not a very good notation since it's trying to imply that the sin function has an inverse, but it doesn't. That's why "sin^{-1}(sin(x)) = x" is not even right in general. The inverse only exists on specific subintervals, and it's also off by a multiple of pi, depending on that subinterval. "arcsin" is then defined as the inverse of sin, restricted to the interval [-pi/2,pi/2]. Of course, the bigger issue here is that f^n for any function f is inherently ambiguous, because it could refer either to the (pointwise) multiplication operation or to the composition operation.

Tomminn 6y ago
This is a very good point. The sine function obviously doesn't have a technical inverse on any interval where it has two values. The notation does make me forget this sometimes. Of course, the situation is different from many other functions without inverses, because the set of all valid inversions can be trivially generated from one solution. Just put a mirror at pi/2 and -pi/2.