91Ó°ÊÓ

To make computations easier, convert the negative angle to a positive coterminal angle in the interval \([0, 2\pi]\). Use the fact that these trigonometric functions are \(2\pi\) periodic:\[-\frac{13 \pi}{12} + 2\pi = -\frac{13 \pi}{12} + \frac{24 \pi}{12} = \frac{11 \pi}{12}\]

Now it's time to calculate the trigonometric values using well-known angles and the properties of these functions. Good angles to remember are: \(\pi/6, \pi/4, \pi/3,\) and their multiples as well as the multiples of \(\pi\).\[\sin\left(\frac{11 \pi}{12}\right) = \sin\left(\pi - \frac{\pi}{12}\right) = \sin\left(\frac{\pi}{12}\right) = \frac{\sqrt{6} - \sqrt{2}}{4}\]\[\cos\left(\frac{11 \pi}{12}\right) = \cos\left(\pi - \frac{\pi}{12}\right) = -\cos\left(\frac{\pi}{12}\right) = -\frac{\sqrt{6} + \sqrt{2}}{4}\]Then, using the definitions of the trigonometric functions, the tangent is given by:\[\tan\left(\frac{11 \pi}{12}\right) = \frac{\sin\left(\frac{11 \pi}{12}\right)}{\cos\left(\frac{11 \pi}{12}\right)} = \frac{\sqrt{6} - \sqrt{2}}{-\sqrt{6} - \sqrt{2}}\]

Finally, simplify the above fraction to obtain the tangent of the given angle:\[\tan\left(\frac{11 \pi}{12}\right) = -1\]