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

User Maximelc
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1 Answer

2 votes

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

User Zajer
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