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.
Are there any natural occurrences of taking a trig function of a trig function?
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0[This](http://dx.doi.org/10.1111/j.1949-8594.1945.tb06335.x) might be of interest. – 2011-09-04
3 Answers
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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1Tiny 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](http://dlmf.nist.gov/11.10). There is also the related Weber function, where the $\cos$ is replaced by $\sin$. – 2011-09-04
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0@J.M.: Thanks. That's a great addendum. – 2011-09-05
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.
Phase modulation would come close.
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0I took a glance at the link. It looks tangential, but I appreciate the info. – 2011-09-03
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0The connection is most clear if you view the signal as a sum of pure tones (via Fourier analysis). – 2011-09-03