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nLab > Latest Changes: tangent function

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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 6th 2023

    Added a few properties off the top of my head. Incidentally, I’m not quite sure what “tangent vector” is doing there under Related Concepts.

    diff, v5, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2023

    The tangent function computes the length of a segment of a tangent to the circle. A tangent vector to the circle is an infinitesimal segment of a tangent to the circle.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 7th 2023

    Cute! If the point on the unit circle is P, at an angle t from the reference point (1, 0), and if a triangle is bounded by three lines, two being radial lines through P and (1, 0) and the third the tangent line at P, then the length of the side included in the tangent line is indeed tan(t). I can see that without even drawing the figure. Do they teach kids that in Germany, I wonder? :-)

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 7th 2023

    Oh, I see now you that put that in, for the congruent triangle bounded by the radial lines but with the tangent at (1, 0) instead (which admittedly is what I had mentally switched over to when I said “I can see that” in my previous comment).

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2023
    • (edited Jun 7th 2023)

    On the one hand I do remember that we looked at these diagrams in school in DE, and I happened to visit a US high-school class a little later where the same structures were introduced without these pictures. Maybe that’s anecdotal evidence for something.

    On the other hand I am hesitant to attribute much to what happened in school, I got my education out of books I read, on this I am with Bertrand Russell here. To be fair, when and where I went to school happened within a rare window in spacetime where things were a little better, but that window has closed again.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 7th 2023

    Excellent quotes from Russell. They remind me of things Noam Chomsky has also said on countless occasions about educational systems (“systems of indoctrination of the young”, as in the first few seconds of this video).

    My own education was “happy” in the sense that I was allowed to be left alone for the most part, so I too have always been largely self-educating, but maybe under circumstances rather different from yours from what I can glean. Conversations for another day, perhaps.

1 to 6 of 6

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