## Trigonometry 7th Edition

$-1$
$\dfrac{3\pi}{2}$ is a quadrantal angle whose terminal side is on negative y-axis with coordinates. The reference angle of any quadrantal angle is $\dfrac{\pi}{2}$. This means that the given angle's reference angle is $\dfrac{\pi}{2}$. Since $\csc{\theta}$ is the reciprocal of the sine function, we first have to find the value of $\sin{\theta}$. $\sin{(\frac{\pi}{2})}$ is a special angle whose value is $1$. Recall that an angle and its reference angle have the same sine value,except possibly in their signs. Since $\dfrac{3\pi}{2}$ is below the x-axis, its sine value is negative. Thus, $\sin{(\frac{3\pi}{2})}=-1$. RECALL: $\csc{\theta} = \dfrac{1}{\sin{\theta}}$ Therefore, $\csc{(\frac{3\pi}{2})} = \dfrac{1}{\sin{(\frac{3\pi}{2}})}=\dfrac{1}{-1} = -1$