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When speaking of arc functions, such as ASIN, ACOS, ATAN, IF it is better to say, "inverse sin, inverse cos, inverse tan; THEN why?"

-Dale-
(04-17-2019 10:29 AM)DrD Wrote: [ -> ]When speaking of arc functions, such as ASIN, ACOS, ATAN, IF it is better to say, "inverse sin, inverse cos, inverse tan; THEN why?"

-Dale-

At school, here in the UK in the 80s, we were taught inverse. Indeed we all were told (and they still are) to get Casio calculators (e.g. fx-80) and they had $$\sin^{-1}$$ printed above the keys. As to why: it was unthinking, being [sort of] the inverse of the sin function.

A more graphical intuition would have been nice, but I don't recall it being taught, nor the reason for the arc reference.
I was always confused (not really, but I found it inconsistent), that $$\sin^{-1}(x)$$ denoted $$\rm{asin(}x)$$, but $$\sin^2(x)$$ denoted $$(\sin(x))^2$$.
(04-17-2019 10:41 PM)ijabbott Wrote: [ -> ]I was always confused (not really, but I found it inconsistent), that $$\sin^{-1}(x)$$ denoted $$\rm{asin(}x)$$, but $$\sin^2(x)$$ denoted $$(\sin(x))^2$$.

The American textbook that I currently use to teach Precalculus uses both $$\sin^{-1}(x)$$ and $$\rm{arcsin(}x)$$, but mostly $$\sin^{-1}(x)$$. When I was a student, our textbook also defined $$\rm{Sin^{-1}(x)}$$ as the inverse function (range from -pi/2 to pi/2) and $$\sin^{-1}(x)$$ as the inverse relation of $$\sin(x)$$ (range from -infinity to infinity).

When discussing the meaning of $$f^{-1}(x)$$, I always feel like I have to apologize to the students for the confusing notation as $$f^{-1}(x)$$ means inverse but $$f^{2}(x)$$ means $$(f(x))^2$$. I sometimes ask them what $$f^{-2}(x)$$ would mean and then say that I don't know either. It could be the square of the inverse or the square of the reciprocal. What if you wanted the reciprocal of the inverse? :-)

I once had a German student who said that she was taught that $$f^{2}(x) = f(f(x))$$. I thought that was silly until I realized that we use this notation for $$d^2/dx^2 (f(x)) = d/dx(d/dx (f(x)))$$.

This was the same student who introduced me to the (not greater than) symbol (also known as ≤ ).
(04-18-2019 03:21 AM)Wes Loewer Wrote: [ -> ]I once had a German student who said that she was taught that $$f^{2}(x) = f(f(x))$$.
I thought that was silly until I realized that we use this notation for $$d^2/dx^2 (f(x)) = d/dx(d/dx (f(x)))$$.

I like the idea ! Think sin as an operator, like D = d/dx

asin sin = 1
asin = sin-1

Valentin cin(x) puzzle, cin(cin(cin(x))) = sin(x)

cin cin cin = sin
cin³ = sin
cin = sin1/3

For calculating accurate cin(x), cin = asinn cin sinn, with big enough n and few Taylor terms of cin
The standard notation in computing is asin/acos/atan. Recent Casio calcs display Asin/Acos/Atan on the keyboard but are still using the confusing sin^-1/etc. notation.
Composition is @ in giac, and iterate composition @@.
The first time, (that I remember seeing), arc-functions called inverse functions, (with the -1 exponent), was on early T.I. calcs. During my basic education classes I remember them referred to as, "arcsin, arccos," etc.

Arc notation, where the angle's function value comes from the arc drawn on the unit circle, seemed more natural to "say," rather than "inverse", which seemed more like a reciprocal thing, in general. In higher education, I mostly encountered them as inverse functions, (instead of arc), which sounded sort of peculiar to me. (Math has always been like a foreign language to me, anyway, so it's not too surprising that I would have this sort of 'dialect' reaction, I guess!).

Thank you, all, for the discussion!

-Dale-
I learned the f^-1 notation from the HP-65:

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