Find the exact value of the given expression using a sum or difference formula. $\sin 285 ^ { \circ }$
A) $\frac { ( \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$
B) $\frac { ( - \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$
C) $\frac { ( \sqrt { 3 } - 1 ) ( 2 \sqrt { 2 } ) } { 8 }$
D) $\frac { ( - \sqrt { 3 } + 1 ) ( 2 \sqrt { 2 } ) } { 8 }$

Question

Quizplus · Accepted Answer

$\sin 285^\circ$ can be rewritten as $\sin(360^\circ - 75^\circ)$, which equals $-\sin 75^\circ$. Using the sum formula for sine, $\sin 75^\circ = \sin(30^\circ + 45^\circ) = \sin 30^\circ \cos 45^\circ + \cos 30^\circ \sin 45^\circ$, which simplifies to $\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$. This equals $\frac{\sqrt{2} + \sqrt{6}}{4}$, and when negated (because we're dealing with $-\sin 75^\circ$), it matches the expression in choice B when simplified.

Find the Exact Value of the Given Expression Using a Sum