If you look at sunrise times at places just south of the Arctic Circle it's pretty obvious that day length is not exactly a sinusoid. See for example Reykjavi

If you look at sunrise times at places just south of the Arctic Circle it's pretty obvious that day length is not exactly a sinusoid. See for example Reykjavik: https://www.timeanddate.com/sun/iceland/reykjavik https://www.timeanddate.com/sun/iceland/reykjavik Ignoring refraction, you have cos(omega_O) = - tan(phi) * tan(delta), where: - omega_0 is the hour angle at sunrise/sunset (basically the time) - phi is the observer's longitude - delta is the sun's declination, which varies over the year. delta is not exactly sinusoidal but that doesn't seem to be the major problem. The hour angle at sunrise is omega_O = arccos(-tan(phi) * tan(delta)) and if delta varies sinusoidally then I think we can wave our hands and say "small angle approximation" to get an approximate sinusoid out the other end, but if tan(phi) gets large enough the approximation breaks down.