Question

Use sum or difference identities to find the exact value of sin285

Answer 1

Angle sum identity:

$\sin285^\circ=\sin(240^\circ+45^\circ)=\sin240^\circ\cos45^\circ+\cos240^\circ\sin45^\circ$

Now

$\sin240^\circ=\sin(180^\circ+60^\circ)=-\sin60^\circ=-\dfrac{\sqrt3}2$

$\cos240^\circ=\cos(180^\circ+60^\circ)=-\cos60^\circ=-\dfrac12$

$\sin45^\circ=\cos45^\circ=\dfrac1{\sqrt2}$

so we end up with

$\sin285^\circ=-\dfrac{\sqrt3}{2\sqrt2}-\dfrac1{2\sqrt2}=-\dfrac{\sqrt3+1}{2\sqrt2}=-\dfrac{\sqrt6+\sqrt2}4$