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. rationalize all denominators.)

Explanation:

Step1: Use the even - function property of cosine

Since \(\cos(-\alpha)=\cos\alpha\), then \(\cos(- 105^{\circ})=\cos(105^{\circ})\).

Step2: Rewrite \(105^{\circ}\) as a sum of two special angles

\(105^{\circ}=60^{\circ}+45^{\circ}\). Then, by the cosine of a sum formula \(\cos(A + B)=\cos A\cos B-\sin A\sin B\), where \(A = 60^{\circ}\) and \(B=45^{\circ}\).
We know that \(\cos60^{\circ}=\frac{1}{2}\), \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\), \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\), and \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\).

Step3: Substitute the values into the formula

$$
LATEXBLOCK0
$$

Answer:

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