Question

Find the exact value of cos(135 degree + 30 degree)

Answer 1

$cos(x+y)=cosxcosy-sinxsiny\\sin(x+y)=sinxcosy+sinycosx\\-------------------------------\\cos(135^o+30^o)=cos135^ocos30^o-sin135^osin30^o=(*)\\----------------------------\\cos135^o=cos(90^o+45^o)=cos90^ocos45^o-sin90^osin45^o\\=0\cdot\frac{\sqrt2}{2}-1\cdot\frac{\sqrt2}{2}=-\frac{\sqrt2}{2}\\\\sin135^o=sin(90^o+45^o)=sin90^ocos45^o+sin45^ocos90^o\\=1\cdot\frac{\sqrt2}{2}+\frac{\sqrt2}{2}\cdot0=\frac{\sqrt2}{2}\\-------------------------------$

$(*)=-\frac{\sqrt2}{2}\cdot\frac{\sqrt3}{2}-\frac{\sqrt2}{2}\cdot\frac{1}{2}=-\frac{\sqrt{6}}{4}-\frac{\sqrt2}{4}=-\frac{\sqrt6+\sqrt2}{4}$