Editing Tangent half-angle formula (section)

==Rational values and Pythagorean triples==
{{main article|Pythagorean triple}}
Starting with a [[Pythagorean triangle]] with side lengths {{mvar|a}}, {{mvar|b}}, and {{mvar|c}} that are positive integers and satisfy {{math|''a''{{sup|2}} + ''b''{{sup|2}} {{=}} ''c''{{sup|2}}}}, it follows immediately that each [[interior angle]] of the triangle has rational values for sine and cosine, because these are just ratios of side lengths.  Thus each of these angles has a rational value for its half-angle tangent, using {{math|tan ''φ''/2 {{=}} sin ''φ'' / (1 + cos ''φ'')}}.

The reverse is also true.  If there are two positive angles that sum to 90°, each with a rational half-angle tangent, and the third angle is a [[right angle]] then a triangle with these interior angles can be [[similar (geometry)|scaled to]] a Pythagorean triangle.  If the third angle is not required to be a right angle, but is the angle that makes the three positive angles sum to 180° then the third angle will necessarily have a rational number for its half-angle tangent when the first two do (using angle addition and subtraction formulas for tangents) and the triangle can be scaled to a [[Heronian triangle]].

Generally, if {{mvar|K}} is a [[Field extension|subfield]] of the complex numbers then {{math|tan ''φ''/2 ∈ ''K'' ∪ {{mset|∞}}}} implies that {{math|{sin ''φ'', cos ''φ'', tan ''φ'', sec ''φ'', csc ''φ'', cot ''φ''} ⊆ ''K'' ∪ {{mset|∞}}}}.