Give the degree measure of θ.
-$$\theta = \csc ^ { - 1 } ( - 2 )$$
A) $- 30 ^ { \circ }$
B) $60 ^ { \circ }$
C) $420 ^ { \circ }$
D) $30 ^ { \circ }$

Question

Quizplus · Accepted Answer

The csc function is the reciprocal of the sin function. Therefore, $\csc(\theta) = -2$ implies $\sin(\theta) = -\frac{1}{2}$. The arcsin or inverse sine of $-\frac{1}{2}$ corresponds to an angle of $-30^\circ$ in the unit circle, which is the same as $330^\circ$ when considering positive angles. However, the range of $\csc^{-1}(x)$ is $90^\circ$ to $270^\circ$ for negative values, meaning we look for an equivalent angle in this range. The correct equivalent angle in the specified range is not directly given, but understanding that $-30^\circ$ is the principal value for the inverse function leads to the selection of $-30^\circ$ as the closest match from the provided options, acknowledging the context of the question might have intended to test conceptual understanding rather than precise range adherence.

Give the Degree Measure of θ θ=csc⁡−1(−2)\theta = \csc ^ { - 1 } ( - 2 )θ=csc−1(−2)

Give the Degree Measure of θ $\theta = \csc ^ { - 1 } ( - 2 )$