4 ms·

I'm sure you used inverse of a cosine multiple times. Didactic math today is just not bothering to give it a name. Probably because people think that sin, cos a

I'm sure you used inverse of a cosine multiple times. Didactic math today is just not bothering to give it a name. Probably because people think that sin, cos and tan is enough. Even ctg which is just inverse of tan is often skipped.

asplake 2y ago
The secant is the reciprocal of a cosine – the hypotenuse over the adjacent
- anyfoo 2y ago
  That’s right, it’s a distribution. And that fact has me, a non-mathematician, personally caused some huge headaches, because I thought I could treat it just like a function… Yeah, turns out really weird things happen if you try to do so without knowing what you’re doing. For example, taking its square does not make sense.
  - grandempire 2y ago
    It is a function. What do you mean?
    - fiddlerwoaroof 2y ago
      The weird thing about 1/cos is it’s discontinuous wherever cos is 0 but, yes, it’s a function.
      - anyfoo 2y ago
        Yeah, that was replied to the wrong comment.
    - anyfoo 2y ago
      Oops, replied to the wrong comment. This is the one I meant to reply to, which is talking about the impulse train, which is not a function: https://news.ycombinator.com/item?id=43741539 https://news.ycombinator.com/item?id=43741539
seanhunter 2y ago
I know what you mean, but as a sibling pointed out for everyone else's benefit, parent is using the word inverse where they mean reciprocal. The inverse of cosine is arccosine (sometimes written acos or cos^{-1}). Secant is the reciprocal of cos ie sec x = 1/cos(x)). Likewise cotan is the reciprocal of tan (1/tan). The inverse of tan is atan/arctan/tan^{-1}. This is confusing for a lot of people because if you write x^{-1} that means 1/x. If you write f^{-1} and f is a function, then _generally_ it means the inverse of f. In the case of trig functions this is doubly confusing because people write sin^2 theta meaning (sin theta)^2 but sin^-1 theta means arcsin theta. That's why in my maths studies they started by teaching you to do the inverse with a -1 so when you see it you don't get confused but changed to preferring arcsin etc as this is unambiguous and if you learn to write this way you won't confuse others.
- Rexxar 2y ago
  It does not help that both reciprocal and inverse come from French, and that their common meanings are reversed in English. I'm not sure whether the meaning of both words has remained constant over time in these two languages, as they both roughly mean "the opposite" and if you want to avoid ambiguity, you simply add context. For example, if you say "inverse function" or "multiplicative inverse" it's not ambiguous. Inverse function: https://en.wikipedia.org/wiki/Inverse_function https://en.wikipedia.org/wiki/Inverse_function / https://fr.wikipedia.org/wiki/Bijection_r%C3%A9ciproque https://fr.wikipedia.org/wiki/Bijection_r%C3%A9ciproque Reciprocal: https://en.wikipedia.org/wiki/Multiplicative_inverse https://en.wikipedia.org/wiki/Multiplicative_inverse / https://fr.wikipedia.org/wiki/Inverse https://fr.wikipedia.org/wiki/Inverse Wikipedia seems to have chosen "multiplicative inverse" over "reciprocal" for title, even though they are clearly indicated as synonymous.
  - seanhunter 2y ago
    That’s a really good point. I will try to remember to do that in future.

asplake 2y ago

The secant is the reciprocal of a cosine – the hypotenuse over the adjacent

anyfoo 2y ago
That’s right, it’s a distribution. And that fact has me, a non-mathematician, personally caused some huge headaches, because I thought I could treat it just like a function… Yeah, turns out really weird things happen if you try to do so without knowing what you’re doing. For example, taking its square does not make sense.
- grandempire 2y ago
  It is a function. What do you mean?
  - fiddlerwoaroof 2y ago
    The weird thing about 1/cos is it’s discontinuous wherever cos is 0 but, yes, it’s a function.
    - anyfoo 2y ago
      Yeah, that was replied to the wrong comment.
  - anyfoo 2y ago
    Oops, replied to the wrong comment. This is the one I meant to reply to, which is talking about the impulse train, which is not a function: https://news.ycombinator.com/item?id=43741539 https://news.ycombinator.com/item?id=43741539

seanhunter 2y ago

I know what you mean, but as a sibling pointed out for everyone else's benefit, parent is using the word inverse where they mean reciprocal. The inverse of cosine is arccosine (sometimes written acos or cos^{-1}). Secant is the reciprocal of cos ie sec x = 1/cos(x)). Likewise cotan is the reciprocal of tan (1/tan). The inverse of tan is atan/arctan/tan^{-1}. This is confusing for a lot of people because if you write x^{-1} that means 1/x. If you write f^{-1} and f is a function, then _generally_ it means the inverse of f. In the case of trig functions this is doubly confusing because people write sin^2 theta meaning (sin theta)^2 but sin^-1 theta means arcsin theta. That's why in my maths studies they started by teaching you to do the inverse with a -1 so when you see it you don't get confused but changed to preferring arcsin etc as this is unambiguous and if you learn to write this way you won't confuse others.

Rexxar 2y ago
It does not help that both reciprocal and inverse come from French, and that their common meanings are reversed in English. I'm not sure whether the meaning of both words has remained constant over time in these two languages, as they both roughly mean "the opposite" and if you want to avoid ambiguity, you simply add context. For example, if you say "inverse function" or "multiplicative inverse" it's not ambiguous. Inverse function: https://en.wikipedia.org/wiki/Inverse_function https://en.wikipedia.org/wiki/Inverse_function / https://fr.wikipedia.org/wiki/Bijection_r%C3%A9ciproque https://fr.wikipedia.org/wiki/Bijection_r%C3%A9ciproque Reciprocal: https://en.wikipedia.org/wiki/Multiplicative_inverse https://en.wikipedia.org/wiki/Multiplicative_inverse / https://fr.wikipedia.org/wiki/Inverse https://fr.wikipedia.org/wiki/Inverse Wikipedia seems to have chosen "multiplicative inverse" over "reciprocal" for title, even though they are clearly indicated as synonymous.
- seanhunter 2y ago
  That’s a really good point. I will try to remember to do that in future.