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(\sqrt2)/(2)-1\cdot(\sqrt2)/(2)=-(\sqrt2)/(2)\\\\sin135^o=sin(90^o+45^o)=sin90^ocos45^o+sin45^ocos90^o\\=1\cdot(\sqrt2)/(2)+(\sqrt2)/(2)\cdot0=(\sqrt2)/(2)\\-------------------------------`


`(*)=-(\sqrt2)/(2)\cdot(\sqrt3)/(2)-(\sqrt2)/(2)\cdot(1)/(2)=-(√(6))/(4)-(\sqrt2)/(4)=-(\sqrt6+\sqrt2)/(4)`