The cosecant function, $\csc(\theta)$, is the reciprocal of the sine function, i.e.,

$\csc(\theta) = \frac{1}{\sin(\theta)}$

We are asked to find $\csc \left( \frac{7\pi}{6} \right)$.

Step 1: Determine $\sin \left( \frac{7\pi}{6} \right)$

The angle $\frac{7\pi}{6}$ is in the third quadrant of the unit circle. In this quadrant, the sine function is negative. The reference angle for $\frac{7\pi}{6}$ is:

$\text{Reference angle} = \frac{7\pi}{6} - \pi = \frac{\pi}{6}$

We know that:

$\sin \left( \frac{\pi}{6} \right) = \frac{1}{2}$

Since $\frac{7\pi}{6}$ is in the third quadrant where sine is negative, we have:

$\sin \left( \frac{7\pi}{6} \right) = -\frac{1}{2}$

Step 2: Find $\csc \left( \frac{7\pi}{6} \right)$

Now, taking the reciprocal of sine:

$\csc \left( \frac{7\pi}{6} \right) = \frac{1}{\sin \left( \frac{7\pi}{6} \right)} = \frac{1}{-\frac{1}{2}} = -2$

Final Answer:

$\csc \left( \frac{7\pi}{6} \right) = -2$

Would you like further details or have any questions about this solution?

Here are five related questions for practice:

1. What is $\csc \left( \frac{5\pi}{6} \right)$?
2. Find the exact value of $\sec \left( \frac{7\pi}{6} \right)$.
3. Calculate $\cot \left( \frac{7\pi}{6} \right)$.
4. What is the sine of $\frac{11\pi}{6}$?
5. Determine the reference angle for $\frac{5\pi}{3}$.

Tip: Always remember that the sine function is negative in the third and fourth quadrants!