Note that sin H < 0 and cos H > 0, a condition that holds only in the 4th quadrant (between 270° and 360°, which is equivalent to being between -90° and 0°). Therefore, H = -33°753 = -02h 15m is the correct answer.

The quadrant ambiguity also arises in computing the azimuth A from (2.4). The quantity cos A may be computed from (2.1), and the signs of sin A and of cos A from Table 2.2 will decide the quadrant. The numerical values of A computed from (2.1) and (2.4) should agree, and computing them both provides a check on the calculation. A difference between the two values indicates either a miscalculation or lack of precision (insufficient number of digits) used in the calculations. The basic point, however, is that the signs of the sine and cosine functions are sufficient to resolve the quadrant question of both H and A. The same remarks hold for any longitudinal-type coordinate that can range from 0° to 360°.

Consider the reverse of our earlier example. Now we are given the latitude, f = 30°, the declination, 8 = -32.263°, and the hour angle, H = -33.753°. Then, from (2.3), we find the altitude, h:

sin h = sin(30) • sin(-32.260°) + cos(30°) • cos(-32.260) • cos(-33.753°) = 0.50000 • (-0.53376) + 0.86603 • 0.84563 • 0.83114 = 0.34202, from which we get h = 20.000°.

Solving (2.4), we can also recover the azimuth:

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