Solar Azimuth Angle (Compass Bearing of the Sun)

The Sun's bearing is the compass direction along the horizon at which it lies, measured clockwise from north: 90 degrees is east, 180 degrees south, 270 degrees west. The formula needs the latitude $\varphi$, the Sun's declination $\delta$ and the hour angle $H$, which is 0 degrees at solar noon, negative in the morning and positive in the afternoon, 15 degrees for each hour. This is spherical trigonometry, not a fit, so it is exact for the angles you give it.

The two-argument arctangent is the heart of it, as in NREL's Solar Position Algorithm (equations 45 and 46). The astronomers' azimuth $\Gamma$ is measured westward from south, and adding half a turn gives the bearing from north. Because atan2 keeps the sign of both of its arguments, it lands in the right quadrant in the morning and in the afternoon, and for a Sun north of the observer at noon in the tropics, which the arccosine form that NOAA's note prints cannot do without a rule about the sign of the hour angle. At latitude 10 degrees north, declination 20 degrees north and $H = 0$, the second argument is $0.17365 - 0.35844 = -0.18479$. It is negative, so the bearing is 0, due north.

The equinox is the easy check. At 45 degrees north the Sun rises due east, stands due south at noon and sets due west: 90, 180 and 270 degrees. At the sunrise hour angle of -90 degrees, $\cos H$ is 0, so the bearing is exactly 90, which is the page's example. In NREL's Golden, Colorado example (17 October 2003, 12:30:30 local time) the report prints 194.34024 degrees; with its declination of -9.316179, hour angle of 11.10629 and latitude of 39.742476 degrees the page returns 194.340265. The gap is the report's own: its hour angle is 0.00002 above what its equation 40 gives, and the azimuth turns 1.26 degrees for each degree of hour angle.

This is the geometric bearing: atmospheric refraction lifts the Sun's apparent height near the horizon but moves its bearing by a negligible amount. Its accuracy is that of the declination and hour angle you give it, so take the declination from the declination page and the hour angle from solar time, which is clock time corrected for the longitude and the equation of time. A bearing is also given when the Sun is below the horizon, and it is undefined only at the zenith or the nadir, where every direction is the same.