Notes: `225^@ = 300^@ - 45^@` Find the exact values of the sine, cosine, and tangent of the angle.

$\displaystyle{\sin{{\left({u}-{v}\right)}}}={\sin{{\left({u}\right)}}}{\cos{{\left({v}\right)}}}-{\cos{{\left({u}\right)}}}{\sin{{\left({v}\right)}}}$

$\displaystyle{\sin{{\left({300}-{45}\right)}}}={\sin{{\left({300}\right)}}}{\cos{{\left({45}\right)}}}-{\cos{{\left({300}\right)}}}{\sin{{\left({45}\right)}}}$

$\displaystyle{\sin{{\left({300}-{45}\right)}}}={\left(-\frac{\sqrt{{3}}}{{2}}\right)}{\left(\frac{\sqrt{{2}}}{{2}}\right)}-{\left(\frac{{1}}{{2}}\right)}{\left(\frac{\sqrt{{2}}}{{2}}\right)}=-\frac{\sqrt{{2}}}{{4}}{\left(\sqrt{{3}}+{1}\right)}$

$\displaystyle{\cos{{\left({u}-{v}\right)}}}={\cos{{\left({u}\right)}}}{\cos{{\left({v}\right)}}}+{\sin{{\left({u}\right)}}}{\sin{{\left({v}\right)}}}$

$\displaystyle{\cos{{\left({300}-{45}\right)}}}={\cos{{\left({300}\right)}}}{\cos{{\left({45}\right)}}}+{\sin{{\left({300}\right)}}}{\sin{{\left({45}\right)}}}$

$\displaystyle{\cos{{\left({300}-{45}\right)}}}={\left(\frac{{1}}{{2}}\right)}{\left(\frac{\sqrt{{2}}}{{2}}\right)}+{\left(-\frac{\sqrt{{3}}}{{2}}\right)}{\left(\frac{\sqrt{{2}}}{{2}}\right)}=\frac{\sqrt{{2}}}{{4}}{\left({1}-\sqrt{{3}}\right)}$

$\displaystyle{\tan{{\left({u}-{v}\right)}}}=\frac{{{\tan{{\left({u}\right)}}}-{\tan{{\left({v}\right)}}}}}{{{1}-{\tan{{\left({u}\right)}}}{\tan{{\left({v}\right)}}}}}$

$\displaystyle{\tan{{\left({300}-{45}\right)}}}=\frac{{{\tan{{\left({300}\right)}}}-{\tan{{\left({45}\right)}}}}}{{{1}-{\tan{{\left({300}\right)}}}{\tan{{\left({45}\right)}}}}}=\frac{{-\sqrt{{3}}-{1}}}{{{1}-\sqrt{{3}}{\left({1}\right)}}}=\frac{{-\sqrt{{3}}-{1}}}{{{1}-\sqrt{{3}}}}$