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In devising challenging exercises for my students, I am tempted to give them something like $\cos(3\sin(4))$, but then I get to wondering whether such a calculation would ever be encountered in practice. Since radians are dimensionless, as are values returned by trig functions, there is no mathematical barrier to this happening, but I was wondering if it ever happened naturally in the course of solving some problem, in mathematics, physics, finance, or elsewhere.

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This does occur. A notable example would be the Bessel function $$J_n(x) = {1 \over \pi}\int_0^{\pi} \cos(nt - x\sin t)\,dt$$ These functions come up in various places in physics and so on. Also, whenever you do a contour integral such as ${\displaystyle \int_{|z| = 1}{1 \over \cos(z)}\,dz}$, if you parameterize the unit circle by $t \rightarrow e^{it}$ you will be doing the integral $$\int_0^{2\pi}{ie^{it} \over \sin(e^{it})}\,dt$$ Due to Euler's formula the denominator is effectively the composition of trigonometric functions. And contour integrals of such functions come up in applications all the time.

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    $\begingroup$ Tiny note: the first integral representation given coincides with a Bessel function only for integer $n$. For general $n$, the function is called an Anger function. There is also the related Weber function, where the $\cos$ is replaced by $\sin$. $\endgroup$ Sep 4 '11 at 0:21
  • $\begingroup$ @J.M.: Thanks. That's a great addendum. $\endgroup$
    – Mike Jones
    Sep 5 '11 at 19:35
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This occurs naturally when you express a plane wave $\mathrm e^{\mathrm i\mathbf k\mathbf x}$ in terms of the angle $\theta$ between $\mathbf k$ and $\mathbf x$ as $\mathrm e^{\mathrm ikx\cos\theta}=\cos(kx\cos\theta)+\mathrm i\sin(kx\cos\theta)$. This is especially relevant when expanding plane waves in terms of cylindrical or spherical waves, which is related to scattering and the Bessel functions mentioned by Zarrax.

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Phase modulation would come close.

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  • $\begingroup$ I took a glance at the link. It looks tangential, but I appreciate the info. $\endgroup$
    – Mike Jones
    Sep 3 '11 at 21:37
  • $\begingroup$ The connection is most clear if you view the signal as a sum of pure tones (via Fourier analysis). $\endgroup$ Sep 3 '11 at 21:47

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