Rewrite $\displaystyle\frac{{{17}\pi}}{{4}}\usepackage{cmbright}\Huge\$ as $\displaystyle{4}\pi+\frac{\pi}{{4}}\usepackage{cmbright}\Huge\$ . Now use the identity $\displaystyle{\sin{{\left(\theta+{2}\pi\right)}}}={\sin{\theta}}\usepackage{cmbright}\Huge\$ to simplify the calculation:

$\displaystyle{\sin{{\left(\frac{{{17}\pi}}{{4}}\right)}}}={\sin{{\left({4}\pi+\frac{\pi}{{4}}\right)}}}={\sin{{\left({2}\pi+\frac{\pi}{{4}}\right)}}}\usepackage{cmbright}\Huge\$

$\displaystyle={\sin{{\left(\frac{\pi}{{4}}\right)}}}={\sin{{\left({{45}}^{\circ}\right)}}}=\frac{\sqrt{{2}}}{{2}}\usepackage{cmbright}\Huge\$