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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: Azimuth Formula Questions
From: George Huxtable
Date: 2005 Oct 22, 16:14 +0100

```Bill wrote-
> In January 2005 there was a discussion you might want to check out under:
>
> Sight Reduction Formula Question
>
> In it George Huxtable et al examine several formulas for finding azimuth.

This is a question that crops up regularly on this list.

Here's the expression to use to obtain azimuth from its Tan. It presumes
that southerly latitudes and declinations are negative, that local hour
angle is measured positively Westwards 0 to 360 degrees, and that azimuths
are measured clockwise from North, 0 to 360 degrees.

az = arctan (sin (hour angle) / (cos (hour angle) sin lat - cos lat tan
dec))

Then to put az into the right quadrant, apply the following rules-

if tan az was negative, add 180 degrees to az.
if hour angle was less than 180 degrees (i.e., to the West of the observer),
add another 180 degrees to az

If a calculator or computer offers a POL or arctan2 function then you don't
even need to apply those rules; the azimuth comes out straightaway in its
correct quadrant, from 0 to 360.
For example, the correct angle results from applying-

POL((tan dec cos lat - cos (hour angle)sin lat, -sin (hour angle)).  Before
applying this formula, however,  check whether the term before the comma and
the term after the comma do not both happen to be zero. If they do, the
angle is indeterminate, and an error may result.

The formula obtaining Az from its tan has the following advantages-

It doesn't need the altitude to have been calculated.

It has no ambiguity, whereas the expression obtaining az from its sin has an
ambiguity about the East-West line that's very awkward to resolve.

It is sensitive over the whole angular range, whereas obtaining az from its
sin is very insensitive and inaccurate near 90 and 270 degrees, and from its
cos, around 0 and 180 degrees.

You can find a similar equation for az in Meeus, "Astronomical Algorithms",
equation 13.5, though there's a slight different in that Meeus, in common
with many astronomers, defines azimuth starting from the South.

It puzzles me why this arctan formula is so little-known and little-used.

George.

```
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