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When examining figure in question, it is important to note that the angle (measured from the central axis) for the light ray in air, θ, is not the angle for the ray in the glass core, which we denote θ′. The law of refraction leads to,

sinθ′=1n1sinθ

assuming nair=1. The angle of incidence for the light ray striking the coating is the complement of θ′, which we denote as θ′comp, and recall that

sinθ′comp=cosθ′=√1−sin2θ′.

In the critical case, θ′comp must equal θc specified by equation n1sinθB=n2sinθr. Therefore,

n2n1=sinθ′comp=√1−sin2θ′=√1−(1n1sinθ)2.

which leads to the result: sinθ=√n21−n22. With n1=1.58 and n2=1.53, we obtain

θ=sin−1(1.582−1.532)=23.20.

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