Answer
$sin\frac{4\pi}{3}=-\frac{\sqrt 3}{2}$,$csc\frac{4\pi}{3}=-\frac{2\sqrt 3}{3}$,$cos\frac{4\pi}{3}=-\frac{1}{2}$, $sec\frac{4\pi}{3}=-2$,$tan\frac{4\pi}{3}=\sqrt 3$,
$cot\frac{4\pi}{3}=\frac{\sqrt 3}{3}$
Work Step by Step
Since, $\frac{4\pi}{3}$ lies on third quadrant, thus its reference angle will be $\frac{4\pi}{3}-\pi-=\frac{\pi}{3}$, which can be considered as common angle to all trigonometric ratios.
The trigonometric ratios and their inverse trigonometric ratios are given as follows:
$sin\frac{4\pi}{3}=-sin\frac{\pi}{3}=-\frac{\sqrt 3}{2}$
$csc\frac{4\pi}{3}=\frac{1}{sin\frac{4\pi}{3}}=-\frac{2\sqrt 3}{3}$
$cos\frac{4\pi}{3}=-cos\frac{\pi}{3}=-\frac{1}{2}$
$sec\frac{4\pi}{3}=\frac{1}{cos\frac{4\pi}{3}}=-2$
$tan\frac{4\pi}{3}=tan\frac{\pi}{3}=\sqrt 3$
$cot\frac{4\pi}{3}=\frac{1}{tan\frac{4\pi}{3}}=\frac{\sqrt 3}{3}$
