Squigonometry by W. Wood (Disclaimer: I Do Not Own This Research Paper)

Squigonometry by W. Wood (Disclaimer: I Do Not Own This Research Paper)

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Squigonometry Author(s): William E. Wood Reviewed work(s): Source: Mathematics Magazine, Vol. 84, No. 4 (October 2011), pp. 257-265 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/10.4169/math.mag.84.4.257 . Accessed: 09/09/2012 06:26
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To develop our theory of squigonometry, we must deﬁne functions that do for squircles what cosine and sine do for circles. If we are to use a coupled initial value problem to deﬁne our functions, the solutions u(t) and v(t) must make the function g(t) = u(t)4 + v(t)4 (2)
constant. We design the following coupled initial value problem with that property in mind:
VOL. 84, NO. 4, OCTOBER 2011
259 3 x (t) = −y(t) 3 y (t) = x(t) x(0) = 1 y(0) = 0 (3)
Thus, if u(t) and v(t) are the solution functions to (3), then g (t) = 4u(t)3 u (t) + 4v(t)3 v (t) = −4u(t)3 v(t)3 + 4v(t)3 u(t)3 = 0, so g(t) must be constant. Since g(0) = 1, it follows that g(t) is identically one, as desired. We denote the unique pair of functions satisfying (3) by cq(t) = x(t) and sq(t) = y(t), the cosquine and squine functions, respectively. Again, note that Theorem 2 is what allows us to use the CIVP as a device to deﬁne these functions. E XERCISE 1. Prove that the cosquine and squine functions are even and odd, respectively. E XERCISE 2. Use a computer algebra system to ﬁnd the Maclaurin series for cosquine and squine. We also deﬁne the tanquent function tq(t) = sq(t) . cq(t)
Then tq(t)4 + 1,
d sq(t)4 + cq(t)4 1 tq(t) = = = 2 dt cq(t) cq(t)2
1 the last equality following from the easily veriﬁable identity tq(t)4 + 1 = cq(t)4 . We see from their graphs in F IGURE 2 that the squigonemetric functions are