Question

find the exact value of the expression given below
\\( \cos \left( 105 ^ { \circ } \
ight) \\)
rewrite the expression using a sum or difference formula. choose the correct answer below
\\( \bigcirc \\) a. \\( \cos \left( 105 ^ { \circ } \
ight) = \cos \left( 60 ^ { \circ } + 45 ^ { \circ } \
ight) = \sin \left( 60 ^ { \circ } \
ight) \cos \left( 45 ^ { \circ } \
ight) - \cos \left( 60 ^ { \circ } \
ight) \sin \left( 45 ^ { \circ } \
ight) \\)
\\( \bigcirc \\) b. \\( \cos \left( 105 ^ { \circ } \
ight) = \cos \left( 60 ^ { \circ } + 45 ^ { \circ } \
ight) = \sin \left( 60 ^ { \circ } \
ight) \cos \left( 45 ^ { \circ } \
ight) + \cos \left( 60 ^ { \circ } \
ight) \sin \left( 45 ^ { \circ } \
ight) \\)
\\( \bigcirc \\) c. \\( \cos \left( 105 ^ { \circ } \
ight) = \cos \left( 60 ^ { \circ } + 45 ^ { \circ } \
ight) = \cos \left( 60 ^ { \circ } \
ight) \cos \left( 45 ^ { \circ } \
ight) + \sin \left( 60 ^ { \circ } \
ight) \sin \left( 45 ^ { \circ } \
ight) \\)
\\( \bigcirc \\) d. \\( \cos \left( 105 ^ { \circ } \
ight) = \cos \left( 60 ^ { \circ } + 45 ^ { \circ } \
ight) = \cos \left( 60 ^ { \circ } \
ight) \cos \left( 45 ^ { \circ } \
ight) - \sin \left( 60 ^ { \circ } \
ight) \sin \left( 45 ^ { \circ } \
ight) \\)
the exact value of \\( \cos \left( 105 ^ { \circ } \
ight) \\) is \\( \square \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize

Explanation:

Step1: Use the cosine sum formula

The formula for \(\cos(A + B)=\cos A\cos B-\sin A\sin B\). Here \(A = 60^{\circ}\), \(B=45^{\circ}\), and \(\cos(105^{\circ})=\cos(60^{\circ}+ 45^{\circ})\). So \(\cos(105^{\circ})=\cos60^{\circ}\cos45^{\circ}-\sin60^{\circ}\sin45^{\circ}\)

Step2: Substitute the values of trigonometric functions

We know that \(\cos60^{\circ}=\frac{1}{2}\), \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\), \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\), \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\)

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Answer:

\(\frac{\sqrt{2}-\sqrt{6}}{4}\)