sin(cos(x)) versus cos(sin(x))

A few days ago I wrote about sufficient conditions for f(g(x)) to bound g(f(x)). This evening I stumbled on an analogous theorem.

For real numbers and ,

cos( sin(x)) > sin( cos(x))

for all real x provided

^2 + ^2 < (/2)^2.

Source: American Mathematical Monthly. February 2009. Solution to problem 11309, page 184.

The reference gives two proofs of the theorem above.

Here's a quick and dirty Python script that suggests the theorem and its converse are both true.

from numpy import *import matplotlib.pyplot as pltN = 200xs = linspace(0, pi, N)ds = linspace(-0.5*pi, 0.5*pi, N)gs = linspace(-0.5*pi, 0.5*pi, N)def f(x, d, g): return cos(g*sin(x)) - sin(d*cos(x))for d in ds: for g in gs: if all(f(xs, d, g) > 0): plt.plot(d, g, 'bo') if d**2 + g**2 > (pi/2)**2: print(d, g) plt.gca().set_aspect("equal")plt.show()

This produces a big blue disk of radius /2, confirming that the condition

^2 + ^2 < (/2)^2

is sufficient. Furthermore, it prints nothing, which suggests the condition is also necessary.

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