Figure 2.5. (a) The horizon system and the hour-angle variant of the equatorial system superposed. The definition of the astronomical triangle for a risen star is illustrated. (b) The equatorial and horizon systems, but now seen from the western side of

the celestial sphere and for a slightly different point of view, for a star at the western horizon. (c) The astronomical triangle extracted from its context on the celestial sphere. Drawings by E.F. Milone.

Table 2.2. Sine and cosine quadrants.

so that H = -33°753 or -33.753/15°/h- -02h 15m = 02h 15m east.

These values make sense because of the location of the star, low in the southern part of the sky. Because the sine function is double valued, i.e., two angles have the same function value: sin Q = sin(180 - Q), there is another mathematical solution for H, however. In the above example, therefore, a possible solution is H = 180° - (-33°753) = 213°753, but this second solution does not make sense physically. The angle is equivalent to 14h 15m west or (noting that 213°753 - 360° = -146°247), -9h 45m, nearly ten hours east of the meridian. It is not possible for a visible star so near the southern horizon to be so far from the celestial meridian at this latitude. So the alternative solution can be ruled out "by inspection" in this case. As a rule, however, the other quadrant solution cannot be dismissed without further calculation. To resolve the question of quadrant, cos H may be computed from (2.5); the actual value need not even be calculated (although a numerical check is always a good idea) because the sign of cos H alone can resolve the ambiguity.

The cosine function is double valued because, cos Q = cos(360 - Q) = cos(-Q), where Q is a given angle, but examination of sin Q resolves the ambiguity. The sine function is non-negative in both the first (0° to 90°) and the second (90° to 180°) quadrants, whereas the cosine function is nonnegative in quadrants one and four (270° - 360°). In quadrant three (180° - 270°), both are negative. Therefore, the quadrant can be determined by the signs of both functions (see Table 2.2).

From the same spherical triangle and trigonometric rules, it is possible to express the transformation from the equatorial to the horizon system:

Table 2.2. Sine and cosine quadrants.


Sign (sin)

Sign (cos)

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