Find the exact value of the trigonometric function. (If an answer is undefined, enter UNDEFINED.)
csc

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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!

Find the exact value of the trigonometric function. (If an answer is undefined, enter UNDEFINED.) csc(7π/6)

Learn how to find the exact value of csc(7π/6) using the unit circle and reciprocal trigonometric functions. The solution involves understanding the sine function in the third quadrant and applying the reference angle theorem. Suitable for students in Grades 10-12.

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