To find \( \csc\left(-\frac{7\pi}{6}\right) \), we proceed as follows:

### Step 1: Recognize the angle
The cosecant function is the reciprocal of sine, so:
\[
\csc(\theta) = \frac{1}{\sin(\theta)}
\]

### Step 2: Simplify the angle
We simplify the given angle \( -\frac{7\pi}{6} \) by adding \( 2\pi \) (since the sine function is periodic with period \( 2\pi \)):

\[
-\frac{7\pi}{6} + 2\pi = -\frac{7\pi}{6} + \frac{12\pi}{6} = \frac{5\pi}{6}
\]

So, \( \csc\left(-\frac{7\pi}{6}\right) = \csc\left(\frac{5\pi}{6}\right) \).

### Step 3: Find the sine of \( \frac{5\pi}{6} \)
The angle \( \frac{5\pi}{6} \) is in the second quadrant, where sine is positive. The reference angle is \( \frac{\pi}{6} \), and we know:
\[
\sin\left(\frac{5\pi}{6}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}
\]

### Step 4: Find cosecant
Since \( \csc(\theta) = \frac{1}{\sin(\theta)} \), we have:
\[
\csc\left(\frac{5\pi}{6}\right) = \frac{1}{\frac{1}{2}} = 2
\]

Thus, \( \csc\left(-\frac{7\pi}{6}\right) = 2 \).

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. What is \( \csc\left(-\frac{\pi}{3}\right) \)?
2. How do you calculate \( \sin\left(-\frac{5\pi}{4}\right) \)?
3. What is the period of the cosecant function?
4. How do you find the reference angle for \( -\frac{7\pi}{6} \)?
5. Can you explain how the reciprocal identities apply to trigonometric functions?

**Tip:** For negative angles, always try to simplify by adding \( 2\pi \) to convert them to positive angles within a standard range.

Learn how to calculate cosec(-7π/6) using angle simplification and the reciprocal identity of trigonometric functions. This step-by-step solution explains how to convert the negative angle into a standard range and find the cosecant value. Suitable for Grades 10-12 students learning advanced trigonometry.

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