Question

find the exact value of the expression given below.
cos (105°)

cos (105°) =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression

Explanation:

Step1: Use the cosine addition formula

The formula for \(\cos(A + B)=\cos A\cos B-\sin A\sin B\). We can write \(105^{\circ}=60^{\circ}+45^{\circ}\). So, \(\cos(105^{\circ})=\cos(60^{\circ}+45^{\circ})\).

Step2: Substitute \(A = 60^{\circ}\) and \(B = 45^{\circ}\) into the formula

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

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Answer:

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