Draw $\displaystyle{{315}}^{\circ}\usepackage{cmbright}\Huge\$ in standard position. The terminal side is in the fourth quadrant and the reference angle is $\displaystyle{{45}}^{\circ}\usepackage{cmbright}\Huge\$ . Use the $\displaystyle{{45}}^{\circ}-{{45}}^{\circ}-{{90}}^{\circ}\usepackage{cmbright}\Huge\$ triangle to find the value of this function at the reference angle:

$\displaystyle{{\sin{{45}}}}^{\circ}=\frac{{1}}{\sqrt{{2}}}=\frac{{1}}{\sqrt{{2}}}\cdot\frac{\sqrt{{2}}}{\sqrt{{2}}}=\frac{\sqrt{{2}}}{{2}}\usepackage{cmbright}\Huge\$

Use the signs of x and y in the fourth quadrant to determine the sign of sine:

$\displaystyle{{\sin{{315}}}}^{\circ}=\frac{{y}}{{1}}=\frac{{{\left(-\right)}}}{{{\left(+\right)}}}={\left(-\right)}\usepackage{cmbright}\Huge\$

So: $\displaystyle{{\sin{{315}}}}^{\circ}=-\frac{\sqrt{{2}}}{{2}}\usepackage{cmbright}\Huge\$