Chapter 7 - Section 7.3 - Double-Angle, Half-Angle, and Product-Sum Formulas - 7.3 Exercises - Page 561: 52

$\frac{\sqrt {26}}{26}$

Given $270^\circ\lt\theta\lt360^\circ$, we have $135^\circ\lt\theta/2\lt180^\circ$ and $sin\frac{\theta}{2}\gt0$ Given $tan\theta=-\frac{5}{12}$, we have $cos\theta=\frac{12}{13}$ and $sin\frac{\theta}{2}=\sqrt {\frac{1-cos\theta}{2}}=\sqrt {\frac{1-12/13}{2}}=\frac{\sqrt {26}}{26}$