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.