To find the exact value of \(\tan(-210^\circ)\), we can use the periodicity of the tangent function and the fact that \(\tan(-\theta) = -\tan(\theta)\).
First, let's convert \(-210^\circ\) to a positive angle by adding \(360^\circ\) (the full rotation):
\[ -210^\circ + 360^\circ = 150^\circ \]
Now we can find:
\[ \tan(-210^\circ) = \tan(150^\circ) \]
Next, we know that \(150^\circ\) is in the second quadrant, where the tangent function is negative. We can find \(\tan(150^\circ)\) by using its reference angle, which is \(180^\circ - 150^\circ = 30^\circ\):
\[ \tan(150^\circ) = -\tan(30^\circ) \]
The value of \(\tan(30^\circ)\) is \(\frac{1}{\sqrt{3}}\), thus:
\[ \tan(150^\circ) = -\frac{1}{\sqrt{3}} \]
Finally, putting this together, we find:
\[ \tan(-210^\circ) = -\tan(150^\circ) = -\left(-\frac{1}{\sqrt{3}}\right) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \]
So, the exact value of \(\tan(-210^\circ)\) is:
\[ \boxed{-\frac{\sqrt{3}}{3}} \]
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