Hello, say you are on the unit sphere and you have 2 rays intersecting it from the origin. You know the spherical coordinates of where these 2 rays intersect the sphere ##(\theta_1,\phi_1),(\theta_2,\phi_2)##.(adsbygoogle = window.adsbygoogle || []).push({});

Now, because we know the dot product of two vectors, it is simple to get that the cosine of the angle between these 2 rays. You convert the spherical coordinates to Cartesian coordinates (with ##r=1##) and take the dot product product to get : $$\cos\theta=\cos\theta_1\cos\theta_2+\sin\theta_1\sin\theta_2(\cos\phi_1\cos\phi_2+\sin\phi_1\sin\phi_2)$$

What I want now is the sine of the angle. Of course, it is easy to do ##\sin\theta=\sin(\cos^{-1}(\cos\theta))##. However, for computational reasons, it would be much more beneficial to me if I could get this sine in terms of sines and cosines of ##\theta_1,\phi_1,\theta_2,\phi_2##. I can take the cross product and do a similar thing to get this, but that formula is horrendous. Because it involves a magnitude, there's a square root of a bunch of squares of a ton of terms. Anyone know what the final result of this computation is? I'm guessing that the terms will simplify significantly, but I wrote them down and aside from a few simple simplifications, could not simplify further.

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# Finding the sine of the angle between 2 rays

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