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

Question

Quizplus · Accepted Answer

To find the exact value of $\sin 285^\circ$, we can express it as a sum or difference of angles whose sine values we know. One way to do this is to express $285^\circ$ as $360^\circ - 75^\circ$ or as $240^\circ + 45^\circ$. Using $360^\circ - 75^\circ$ simplifies the calculation since $\sin(360^\circ - \theta) = -\sin(\theta)$. Thus, $\sin 285^\circ = -\sin 75^\circ$. The angle $75^\circ$ can be expressed as $30^\circ + 45^\circ$. Using the sum formula for sine, $\sin(a + b) = \sin a \cos b + \cos a \sin b$, we get:$$\sin 75^\circ = \sin(30^\circ + 45^\circ) = \sin 30^\circ \cos 45^\circ + \cos 30^\circ \sin 45^\circ$$Given that $\sin 30^\circ = \frac{1}{2}$, $\cos 45^\circ = \frac{\sqrt{2}}{2}$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$, and $\sin 45^\circ = \frac{\sqrt{2}}{2}$, substituting these values gives:$$\sin 75^\circ = \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) + \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} = \frac{\sqrt{2} + \sqrt{6}}{4}$$Simplifying further, we get:$$\sin 75^\circ = \frac{\sqrt{2}(1 + \sqrt{3})}{4} = \frac{1 + \sqrt{3}}{2\sqrt{2}}$$Therefore, $\sin 285^\circ = -\sin 75^\circ = -\frac{1 + \sqrt{3}}{2\sqrt{2}}$, which matches option C: $\frac{-\sqrt{3} - 1}{2\sqrt{2}}$.

Find the Exact Value of the Given Expression Using a Sum