Answer
$\displaystyle \sin\theta=-\frac{1}{2}$
$\displaystyle \cos\theta=-\frac{\sqrt{3}}{2}$
$\displaystyle \tan\theta=\frac{\sqrt{3}}{3}$
$\cot\theta=\sqrt{3}$
$\displaystyle \sec\theta=-\frac{2\sqrt{3}}{3}$
$\csc\theta=-2$
Work Step by Step
For any real number $s$ represented by a directed arc on the unit circle,
$\sin s=y\quad \cos s=x \quad \displaystyle \tan s=\frac{y}{x} (x\neq 0)$
$\displaystyle \csc s=\frac{1}{y} (y\neq 0)\quad \displaystyle \sec s=\frac{1}{x} (x\neq 0) \displaystyle \quad\cot s=\frac{x}{y} (y\neq 0)$.
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$\displaystyle \sin\theta=y=-\frac{1}{2}$
$\displaystyle \cos\theta=x=-\frac{\sqrt{3}}{2}$
$\displaystyle \tan\theta=\frac{y}{x}=\frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}}=-\frac{1}{2}(-\frac{2}{\sqrt{3}})=\frac{1}{\sqrt{3}}$
$\displaystyle =\frac{1}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{3}}{3}$
$\displaystyle \cot\theta=\frac{x}{y}=\frac{-\frac{\sqrt{3}}{2}}{-\frac{1}{2}}=-\frac{\sqrt{3}}{2}(-\frac{2}{1})=\sqrt{3}$
$\displaystyle \sec\theta=\frac{1}{x}=\frac{1}{-\frac{\sqrt{3}}{2}}=-\frac{2}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}}=-\frac{2\sqrt{3}}{3}$
$\displaystyle \csc\theta=\frac{1}{y}=\frac{1}{-\frac{1}{2}}=-2$
