Are there any natural occurrences of taking a trig function of a trig function?

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.

Solution 1:

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.

Solution 2:

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.

Solution 3:

Phase modulation would come close.