Find the exact values of $ an( heta) $ for $ heta $ on the unit circle at each 30-degree increment, and explain the symmetry properties of the tangent function on the unit circle.

For each 30-degree increment ($ \theta $) on the unit circle, we have:

$ \tan(0^\circ) = 0 $

$ \tan(30^\circ) = \frac{1}{\sqrt{3}} $

$ \tan(60^\circ) = \sqrt{3} $

$ \tan(90^\circ) = \text{undefined} $

$ \tan(120^\circ) = -\sqrt{3} $

$ \tan(150^\circ) = -\frac{1}{\sqrt{3}} $

$ \tan(180^\circ) = 0 $

$ \tan(210^\circ) = \frac{1}{\sqrt{3}} $

$ \tan(240^\circ) = \sqrt{3} $

$ \tan(270^\circ) = \text{undefined} $

$ \tan(300^\circ) = -\sqrt{3} $

$ \tan(330^\circ) = -\frac{1}{\sqrt{3}} $

$ \tan(360^\circ) = 0 $

The tangent function is periodic with a period of $ 180^\circ $, hence $ \tan(\theta + 180^\circ) = \tan(\theta) $.

For each 30-degree increment ($ heta $) on the unit circle:

$ an(0^circ) = 0 $

$ an(30^circ) = frac{1}{sqrt{3}} $

$ an(60^circ) = sqrt{3} $

$ an(90^circ) = ext{undefined} $

$ an(120^circ) = -sqrt{3} $

$ an(150^circ) = -frac{1}{sqrt{3}} $

The tangent function has the property that $ an( heta) = an(180^circ + heta) $.

For 30-degree increments ($ heta $) on the unit circle:

$ an(0^circ) = 0 $

$ an(30^circ) = frac{1}{sqrt{3}} $

$ an(60^circ) = sqrt{3} $

$ an( heta + 180^circ) = an( heta) $.

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