Use a sum or difference identity to find the exact value.
-$\sin 255 ^ { \circ }$
A) $\frac { - \sqrt { 6 } + \sqrt { 2 } } { 4 }$
B) $\frac { - \sqrt { 6 } - \sqrt { 2 } } { 4 }$
C) $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$
D) $\frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$

Question

Quizplus · Accepted Answer

$\sin 255^\circ$ can be rewritten using the sum identity as $\sin(270^\circ - 15^\circ)$. Using the identity $\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$, we get $\sin 270^\circ \cos 15^\circ - \cos 270^\circ \sin 15^\circ$. Since $\sin 270^\circ = -1$ and $\cos 270^\circ = 0$, the expression simplifies to $-\cos 15^\circ$. Using the half-angle formula, $\cos 15^\circ = \sqrt{\frac{1 + \cos 30^\circ}{2}}$, and since $\cos 30^\circ = \sqrt{3}/2$, we find $\cos 15^\circ = \sqrt{\frac{1 + \sqrt{3}/2}{2}}$. Simplifying further, we get $\cos 15^\circ = \frac{\sqrt{6} + \sqrt{2}}{4}$, so $-\cos 15^\circ = \frac{-\sqrt{6} - \sqrt{2}}{4}$, which matches choice B.

Use a Sum or Difference Identity to Find the Exact sin⁡255∘\sin 255 ^ { \circ }sin255∘

Use a Sum or Difference Identity to Find the Exact $\sin 255 ^ { \circ }$