What is the value of $\tan \left(-\frac{5 \pi}{4}\right)$ to the nearest ten-thousandth?

Step 1 ：The tangent of an angle in the unit circle is defined as the ratio of the y-coordinate to the x-coordinate of the point on the unit circle corresponding to that angle.

Step 2 ：The angle \(-\frac{5 \pi}{4}\) is equivalent to \(\frac{3 \pi}{4}\) in the unit circle, which is in the second quadrant.

Step 3 ：In the second quadrant, the x-coordinate is negative and the y-coordinate is positive, so the tangent of the angle is negative.

Step 4 ：We can calculate the exact value using the numpy library in Python.

Step 5 ：The calculated value of the tangent is approximately -0.9999999999999997.

Step 6 ：Rounding this value to the nearest ten-thousandth gives -1.0000.

Step 7 ：Final Answer: The value of \(\tan \left(-\frac{5 \pi}{4}\right)\) to the nearest ten-thousandth is \(\boxed{-1.0000}\).