Question

Evaluate sin105° using the sum and differences identities underdstanding 105°=45°+60°

Answer 1

$\bf \textit{Sum and Difference Identity} \\\\ sin(\alpha + \beta)=sin(\alpha)cos(\beta) + cos(\alpha)sin(\beta)\\\\ -------------------------------\\\\ sin(45^o+60^o)\implies sin(45^o)cos(60^o)+cos(45^o)sin(60^o) \\\\\\ \cfrac{\sqrt{2}}{2}\cdot \cfrac{1}{2}+\cfrac{\sqrt{2}}{2}\cdot \cfrac{\sqrt{3}}{2}\implies \cfrac{\sqrt{2}}{4}+\cfrac{\sqrt{6}}{4}\implies \cfrac{\sqrt{2}+\sqrt{6}}{4}$

check your Unit Circle for those angles.