Trigonometric Functions as Projections

The period of the tangent function

1. The domain of \(\tan(\theta)\) is \(\theta \in (-\infty,+\infty), \theta \neq 90^\circ + n\times 180^\circ;\quad n\in\mathbb(Z)\).

   (It is necessary to specifically exclude the values where \(\tan(\theta)\) is undefined.)

2. \(\tan(360\,000\,036^\circ) = \tan(180^\circ\times 2\,000\,000 + 36^\circ) = \tan(36^\circ)\) and
   \(\tan(216^\circ) = \tan(180^\circ + 36^\circ) = \tan(36^\circ)\) so
   \(\tan(360\,000\,036^\circ)-\tan(216^\circ) = \tan(36^\circ) - \tan(36^\circ) = 0\).
3. 1. \(\tan(\theta)=0\) for \(\theta = n\times 180^\circ\) for integer \(n\). That is, \(\theta \in \{ \ldots -360^\circ, -180^\circ, 0^\circ, 180^\circ, 360^\circ, \ldots \}\)
   2. \(\tan(\theta) = 1\) for \(\theta = 45^\circ + n\times 180^\circ\) for integer \(n\). That is, \(\theta \in \{ \ldots -315^\circ, -135^\circ, 45^\circ, 225^\circ, 405^\circ, \ldots \}\)
   3. \(\tan(\theta) = -1\) for \(\theta = -45^\circ + n\times 180^\circ\) for integer \(n\). That is, \(\theta\in \{ \ldots -405^\circ, -225^\circ, -45^\circ, 135^\circ, 315^\circ, \ldots \}\)

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