Evaluate the expression without using a calculator. \( \operatorname{arccsc}(2) \)

To evaluate \( \operatorname{arccsc}(2) \), we start by recalling that the cosecant function is the reciprocal of sine. Therefore, \( \operatorname{arccsc}(2) \) is asking for the angle whose cosecant is 2. This means we need to find an angle \( \theta \) such that: \[ \csc(\theta) = 2 \] Since cosecant is the reciprocal of sine, we can rewrite this as: \[ \sin(\theta) = \frac{1}{2} \] The basic sine value of \( \frac{1}{2} \) occurs at angles in the unit circle. We know that: \[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \] Thus, the principal value (the angle in the range of \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \)) that corresponds to \( \sin(\theta) = \frac{1}{2} \) is: \[ \theta = \frac{\pi}{6} \] Hence, we have: \[ \operatorname{arccsc}(2) = \frac{\pi}{6} \]