Answer
$sin\frac{5\pi}{6}=\frac{1}{2}$
$cos\frac{5\pi}{6}=-\frac{\sqrt 3}{2}$
$tan\frac{5\pi}{6}=-\frac{\sqrt 3}{3}$
$csc\frac{5\pi}{6}=2$
$sec\frac{5\pi}{6}=-\frac{2\sqrt 3}{ 3}$
$cot\frac{5\pi}{6}=-\sqrt 3$
Work Step by Step
Since, $\frac{5\pi}{6}$ lies on second quadrant, thus its reference angle will be $\pi-\frac{5\pi}{6}=\frac{\pi}{6}$, which can be considered as common angle to all trigonometric ratios.
The trigonometric ratios and their inverse trigonometric ratios are given as follows:
$sin\frac{5\pi}{6}=sin\frac{\pi}{6}=\frac{1}{2}$
$cos\frac{5\pi}{6}=-cos\frac{\pi}{6}=-\frac{\sqrt 3}{2}$
$tan\frac{5\pi}{6}=-tan\frac{\pi}{6}=-\frac{\sqrt 3}{3}$
$csc\frac{5\pi}{6}=\frac{1}{sin\frac{5\pi}{6}}=2$
$sec\frac{5\pi}{6}=\frac{1}{cos\frac{5\pi}{6}}=-\frac{2\sqrt 3}{ 3}$
$cot\frac{5\pi}{6}=\frac{1}{tan\frac{5\pi}{6}}=-\sqrt 3$
