Sine of the times.

[Wes Loewer]

(06-12-2024 08:58 PM)Nigel (UK) Wrote:  Gauss found this odious, because he felt that \(\sin^2(x)\) should mean \(\sin(\sin(x))\) (and he has a point!) but since the repeated sine is used so rarely this isn't an issue in practice.

Nigel (UK)

(Added) Just to confuse things, \(\sin^{-1}x\) means the inverse sine of \(x\), not \(1/\sin(x)\)!

Some years ago I had a German student who told me that in Germany, they are taught that \(f^2(x)\) means \(f(f(x))\). When I asked her about \(\sin^2(x)\) she said that trig functions were special exceptions.

I thought this was rather odd until it occurred to me that this is exactly what we do with derivatives: \({d^2\over dx^2} f(x)\) means \({d\over dx}{d\over dx} f(x)\). The point you make about \(\sin^{-1}x\) is consistent with this notation.


The nth derivative is often denoted \(f^{(n)}(x)\) with the n in parentheses. Using this notation, I have seen \(f^{(0)}(x)\) (the zeroth derivative) to mean \(f(x)\) and \(f^{(-1)}(x)\) (the inverse derivative) to mean the antiderivative of \(f(x)\).

[Wes Loewer]

(06-12-2024 08:58 PM)Nigel (UK) Wrote:  (Added) Just to confuse things, \(\sin^{-1}x\) means the inverse sine of \(x\), not \(1/\sin(x)\)!

I tell my students, whatever you do, don't ever write \(\sin^{-2}x\).
Possible meanings:

• \(\sin^{-1}(\sin^{-1}(x))\)
• \((\sin^{-1}(x))^2\)
• \(1 \over \sin^2(x) \)