$$\text { Find all values of } \theta \text {, if } \theta \text { is in the interval } \left[ 0,360 ^ { \circ } \right) \text { and has the given function value. }$$
-$\csc \theta = - \frac { 2 \sqrt { 3 } } { 3 }$

Question

Quizplus · Accepted Answer

The cosecant function, $\csc \theta = - \frac { 2 \sqrt { 3 } } { 3 }$, is the reciprocal of the sine function. The given value corresponds to $\sin \theta = - \frac { \sqrt { 3 } } { 2 }$. This sine value occurs in the third and fourth quadrants where sine is negative. The reference angle for $\sin \theta = \frac { \sqrt { 3 } } { 2 }$ is $60^\circ$. Therefore, the angles in the interval $[0,360^\circ)$ that have this sine value are $180^\circ + 60^\circ = 240^\circ$ and $360^\circ - 60^\circ = 300^\circ$.

- csc⁡θ=−233\csc \theta = - \frac { 2 \sqrt { 3 } } { 3 }cscθ=−323​​

- $\csc \theta = - \frac { 2 \sqrt { 3 } } { 3 }$