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Abstract
Two questions any student can ask — Where does the Sun rise? and where exactly is the Sun right now, and how can I calculate it? — have exact answers that fit inside Class-XII mathematics. We take the solar-position equations published by the U.S. National Oceanic and Atmospheric Administration (NOAA), a verified source whose formulae are quoted here verbatim with their provenance, and reduce them with nothing more than the syllabus tools of trigonometry, inverse trigonometric functions and one spherical-triangle identity. The reduction produces a single general equation for the Sun’s azimuth,
$$ A = 180^\circ + \tan^{-1} \left[ \frac{\sin H} {\cos H \sin\phi - \tan\delta \cos\phi} \right]. $$
which needs no separate morning/afternoon cases and is valid at every hour, in every season. Setting the zenith distance to the standard sunrise value $$ z_0 = 90.833^\circ $$ gives the classroom sunrise formula $$ A_0 = \cos^{-1} \left[ \frac{\sin\delta - \sin\phi\cos z_0} {\cos\phi\sin z_0} \right]. $$. Every formula is validated against NOAA’s own operational algorithm, which we execute exactly as published.