Answer
$-2$
Work Step by Step
RECALL:
(1)The reference angle of a given angle is equal to the smallest acute angle that the terminal side makes with the x-axis.
(2) Based on the location of the terminal side of an angle $\theta$, the reference angle can be found using the formula:
(i) Quadrant I: $\theta$
(ii) Quadrant II: $\pi-\theta$
(iii) Quadrant III: $\theta-\pi$
(iv) Quadrant IV: $2\pi-\theta$
The given angle is in Quadrant III so its reference angle is:
$=\dfrac{7\pi}{6}-\pi
\\=\dfrac{\pi}{6}$
The value of cosecant in Quadrant III is negative.
Thus,
$\csc{\frac{7\pi}{6}} = -\csc{\frac{\pi}{6}}$
RECALL:
$\csc{\theta} = \dfrac{1}{\sin{\theta}}$
Thus,
$-\csc{\frac{\pi}{6}} = -\dfrac{1}{\sin{\frac{\pi}{6}}}$
$\dfrac{\pi}{6}$ is a special angle whose sine value is $0.5$.
Thus,
$\csc{\dfrac{7\pi}{6}}=-\dfrac{1}{0.5}=-2$
