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=-\frac{\sqrt3}2`

`\cos240^\circ=\cos(180^\circ+60^\circ)=-\cos60^\circ=-\frac12`

`\sin45^\circ=\cos45^\circ=\frac1{\sqrt2}`

so we end up with

`\sin285^\circ=-(\sqrt3)/(2\sqrt2)-\frac1{2\sqrt2}=-(\sqrt3+1)/(2\sqrt2)=-\frac{\sqrt6+\sqrt2}4`