Answer
$sin~165°=\frac{\sqrt {2-\sqrt 3}}{2}$
$cos~165°=-\frac{\sqrt {2+\sqrt 3}}{2}$
$tan~165°=-2+\sqrt 3$
Work Step by Step
$165°$ lies in the second quadrant.
$sin~165°=sin\frac{330°}{2}=+\sqrt {\frac{1-cos~330°}{2}}=\sqrt {\frac{1-\frac{\sqrt 3}{2}}{2}}=\sqrt {\frac{2-\sqrt 3}{4}}=\frac{\sqrt {2-\sqrt 3}}{2}$
$cos~165°=cos\frac{330°}{2}=-\sqrt {\frac{1+cos~330°}{2}}=-\sqrt {\frac{1+\frac{\sqrt 3}{2}}{2}}=-\sqrt {\frac{2+\sqrt 3}{4}}=-\frac{\sqrt {2+\sqrt 3}}{2}$
$tan~165°=tan\frac{330°}{2}=\frac{1-cos~330°}{sin~330°}=\frac{1-\frac{\sqrt 3}{2}}{-\frac{1}{2}}=-2+\sqrt 3$
