$Find the value of tan(frac{4pi}{3}) using the unit circle$

$To \ find \ the \ value \ of \ tan(\frac{4\pi}{3}) \ using \ the \ unit \ circle, \ we \ first \ need \ to \ determine \ the \ reference \ angle. \ $

$The \ angle \ \frac{4\pi}{3} \ is \ in \ the \ third \ quadrant, \ and \ its \ reference \ angle \ is \ \frac{4\pi}{3} – \pi = \frac{\pi}{3}. \ $

$Using \ the \ unit \ circle, \ the \ coordinates \ for \ \frac{\pi}{3} \ are \ (\frac{1}{2}, \ \frac{\sqrt{3}}{2}). \ $

$Since \ the \ angle \ \frac{4\pi}{3} \ is \ in \ the \ third \ quadrant, \ both \ x \ and \ y \ coordinates \ are \ negative: \ (-\frac{1}{2}, \ -\frac{\sqrt{3}}{2}). \ $

$Finally, \ the \ tangent \ is \ the \ ratio \ of \ y \ to \ x: \ tan(\frac{4\pi}{3}) = \frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = \sqrt{3}. \ $

$Therefore, \ tan(\frac{4\pi}{3}) = \sqrt{3}. \ $

$To find tan(frac{4pi}{3}), we use the unit circle approach. $

$The angle frac{4pi}{3} lies in the third quadrant, and the reference angle is frac{pi}{3}. $

$In the third quadrant, the coordinates for frac{pi}{3} are (-frac{1}{2}, -frac{sqrt{3}}{2}). $

$The tangent function in the third quadrant is positive, so we compute tan(frac{4pi}{3}): frac{-frac{sqrt{3}}{2}}{-frac{1}{2}} = sqrt{3}. $

$Thus, tan(frac{4pi}{3}) = sqrt{3}. $

$Find tan(frac{4pi}{3}) using the unit circle. $

$frac{4pi}{3} is in the third quadrant. $

$Reference angle is frac{pi}{3}. $

$Coordinates: (-frac{1}{2}, -frac{sqrt{3}}{2}). $

$tan(frac{4pi}{3}) = sqrt{3}. $

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