I ran throughout an previous article [1] that gave a type of multiplication desk for trig capabilities and inverse trig capabilities. Right here’s my model of the desk.
I made a number of modifications from the unique. First, I used LaTeX, which didn’t exist when the article was written in 1957. Second, I solely embrace sin, cos, and tan; the unique additionally included csc, sec, and cot. Third, I reversed the labels of the rows and columns. Every cell represents a trig operate utilized to an inverse trig operate.
The third level requires a little bit elaboration. The desk represents operate composition, not multiplication, however is expressed within the format of a multiplication desk. For the composition f( g(x) ), do you count on f to be on the aspect or prime? It wouldn’t matter if the capabilities commuted beneath composition, however they don’t. I feel it feels extra standard to place the outer operate on the aspect; the writer make the other alternative.
The identities within the desk are all straightforward to show, so the outcomes aren’t attention-grabbing a lot because the association. I’d by no means seen these identities organized right into a desk earlier than. The matrix of identities just isn’t symmetric, however the 2 by 2 matrix within the higher left nook is as a result of
sin(cos−1(x)) = cos(sin−1(x)).
The entries of the third row and third column are usually not symmetric, although they do have some similarities.
You may show the identities within the sin, cos, and tan rows by specializing in the angles θ, φ, and ψ under respectively as a result of θ = sin−1(x), φ = cos−1(x), and ψ = tan−1(x). This exhibits that the sq. roots within the desk above all fall out of the Pythagorean theorem.
See the subsequent publish for the hyperbolic analog of the desk above.
[1] G. A. Baker. Multiplication Tables for Trigonometric Operators. The American Mathematical Month-to-month, Vol. 64, No. 7 (Aug. – Sep., 1957), pp. 502–503.
