Find the exact value of the expression. Write the answer as a single fraction. Do not use a calculator.
[tex]sin \frac{23 \pi }{4} cos \frac{11 \pi }{6} + cos \frac{23 \pi }{4} sin \frac{11 \pi }{6} [/tex]
Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression. Rationalize any denominators.