In analogy with a spherical (spatial) triangle there are only three fundamental loop equations for any spherical (spatial) polygon, namely the real (dual) sine, sine-cosine, and cosine laws. All other loop equations which fall naturally into groups of subsidiary sine, sine-cosine, and cosine laws can be derived from the three basic laws. Novel half-tangent laws are derived which are expressions for the common root of pairs of sine and sine-cosine laws.

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