`-165^@` Find the exact values of the sine, cosine, and tangent of the angle.

$\displaystyle-{{165}}^{\circ}=-{\left({{120}}^{\circ}+{{45}}^{\circ}\right)}$

$\displaystyle{\sin{{\left(-{165}\right)}}}=-{\sin{{\left({165}\right)}}}$

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

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

$\displaystyle{\left[{\sin{{\left(-{\left({120}+{45}\right)}\right)}}}\right]}=-{\left[{\sin{{\left({120}\right)}}}{\cos{{\left({45}\right)}}}+{\cos{{\left({120}\right)}}}{\sin{{\left({45}\right)}}}\right]}$

$\displaystyle{\left[{\sin{{\left(-{\left({120}+{45}\right)}\right)}}}\right]}=-{\left[{\left(\frac{\sqrt{{3}}}{{2}}\right)}{\left(\frac{\sqrt{{2}}}{{2}}\right)}+{\left(-\frac{{1}}{{2}}\right)}{\left(\frac{\sqrt{{2}}}{{2}}\right)}\right]}=-\frac{\sqrt{{2}}}{{4}}{\left(\sqrt{{3}}+{1}\right)}$

$\displaystyle{\cos{{\left(-{165}\right)}}}={\cos{{\left({165}\right)}}}$

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

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

$\displaystyle{\cos{{\left(-{\left({120}+{45}\right)}\right)}}}={\cos{{\left({120}\right)}}}{\cos{{\left({45}\right)}}}-{\sin{{\left({120}\right)}}}{\sin{{\left({45}\right)}}}$

$\displaystyle{\cos{{\left(-{\left({120}+{45}\right)}\right)}}}={\left(-\frac{{1}}{{2}}\right)}{\left(\frac{\sqrt{{2}}}{{2}}\right)}-{\left(\frac{\sqrt{{3}}}{{2}}\right)}{\left(\frac{\sqrt{{2}}}{{2}}\right)}=-\frac{\sqrt{{2}}}{{4}}{\left({1}+\sqrt{{3}}\right)}$

$\displaystyle{\tan{{\left(-{165}\right)}}}=-{\tan{{\left({165}\right)}}}$

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

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

$\displaystyle-{\tan{{\left({120}+{45}\right)}}}=-{\left[\frac{{{\tan{{\left({120}\right)}}}+{\tan{{\left({45}\right)}}}}}{{{1}-{\tan{{\left({120}\right)}}}{\tan{{\left({45}\right)}}}}}\right]}=-{\left[\frac{{-\sqrt{{3}}+{1}}}{{{1}-{\left(-\sqrt{{3}}\right)}{\left({1}\right)}}}\right]}=\frac{{\sqrt{{3}}-{1}}}{{{1}+\sqrt{{3}}}}$