Find the exact value of the given expression. $$\cos \left( 120 ^ { \circ } + 315 ^ { \circ } \right)$$
A) $\frac { ( 1 - \sqrt { 3 } ) ( 2 \sqrt { 2 } ) } { 8 }$
B) $\frac { ( - 1 + \sqrt { 3 } ) ( 2 \sqrt { 2 } ) } { 8 }$
C) $\frac { ( 1 + \sqrt { 3 } ) ( 2 \sqrt { 2 } ) } { 8 }$
D) $\frac { ( - 1 - \sqrt { 3 } ) ( 2 \sqrt { 2 } ) } { 8 }$

Question

Quizplus · Accepted Answer

First, add the angles: $120^\circ + 315^\circ = 435^\circ$. Since the cosine function is periodic with a period of $360^\circ$, $\cos(435^\circ) = \cos(435^\circ - 360^\circ) = \cos(75^\circ)$. The exact value of $\cos(75^\circ)$ can be found using the cosine of a sum formula: $\cos(75^\circ) = \cos(45^\circ + 30^\circ) = \cos(45^\circ)\cos(30^\circ) - \sin(45^\circ)\sin(30^\circ)$. Substituting the exact values, we get $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} - \sqrt{2}}{4}$, which simplifies to $\frac{(\sqrt{3} - 1)(\sqrt{2})}{4}$ or $\frac{(-1 + \sqrt{3})(2\sqrt{2})}{8}$, matching choice B.

Find the Exact Value of the Given Expression cos⁡(120∘+315∘)\cos \left( 120 ^ { \circ } + 315 ^ { \circ } \right)cos(120∘+315∘)

Find the Exact Value of the Given Expression $\cos \left( 120 ^ { \circ } + 315 ^ { \circ } \right)$