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Some Useful Groups in the Teaching of Elementary Trigonometry

Published online by Cambridge University Press:  03 November 2016

G. A. Miller
G. A. Miller*
Affiliation:
University of Illinois
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Extract

The operation (c) of taking the complement of an angle (α) is of order (period) two, since 90° –(90° –α) = α. Similarly, the operation (s) of taking the supplement of an angle is of order two. If we perform these two operations in succession there results an operation (cs), which is of order four since it changes α into α + 90° Hence the group {c, s} generated by c and s is the dihedral rotation group of order 8. This fact follows directly from the general theorem, Two operations of order two generate the dihedral rotation group whose order is twice the order of their product.

Type
Research Article
Information
The Mathematical Gazette , Volume 3 , Issue 59 , October 1906 , pp. 353 - 357
Copyright
Copyright © Mathematical Association 1906

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References

Page 353 of note * Bulletin of the American Mathematical Society, vol. 7 (1901), p. 424.

Page 354 of note * Quarterly Journal of Mathematics, vol. 37 (1905), p. 82.

Page 354 of note † Cf. Quarterly Journal of Mathematics, vol 37, (1906), p. 228.

Page 355 of note * We could not use only one fundamental angle since the octic group is non-cyclic.

Page 356 of note * Cf. Annals of Mathematics, vol. 6 (1905), p. 41.

Page 356 of note † Since t is one of the assumed generators of our group of order 16 the functions of the angle (45° − a) which corresponds to t are supposed to be known.