find the exact value of the expression \\( \\sin 225 ^ { \\circ } \\cdot \\cos 105 ^ { \\circ } \\).\n\\(…

find the exact value of the expression \\( \\sin 225 ^ { \\circ } \\cdot \\cos 105 ^ { \\circ } \\).\n\\( \\sin 225 ^ { \\circ } \\cdot \\cos 105 ^ { \\circ } = \\) \n(simplify your answer, including any radicals. use integers or fractions far \nnumbers in the expression.)

find the exact value of the expression \\( \\sin 225 ^ { \\circ } \\cdot \\cos 105 ^ { \\circ } \\).\n\\( \\sin 225 ^ { \\circ } \\cdot \\cos 105 ^ { \\circ } = \\) \n(simplify your answer, including any radicals. use integers or fractions far \nnumbers in the expression.)

Answer

Explanation:

Step1: Use the product - to - sum formula

The product - to - sum formula is (\sin A\cos B=\frac{1}{2}[\sin(A + B)+\sin(A - B)]). Here (A = 225^{\circ}) and (B=105^{\circ}), so (\sin225^{\circ}\cos105^{\circ}=\frac{1}{2}[\sin(225^{\circ}+105^{\circ})+\sin(225^{\circ}-105^{\circ})]).

Step2: Calculate (A + B) and (A - B)

(A + B=225^{\circ}+105^{\circ}=330^{\circ}), (A - B=225^{\circ}-105^{\circ}=120^{\circ}). Then (\sin225^{\circ}\cos105^{\circ}=\frac{1}{2}[\sin330^{\circ}+\sin120^{\circ}]).

Step3: Find the values of (\sin330^{\circ}) and (\sin120^{\circ})

We know that (\sin330^{\circ}=\sin(360^{\circ}-30^{\circ})=-\sin30^{\circ}=-\frac{1}{2}), (\sin120^{\circ}=\sin(180^{\circ}-60^{\circ})=\sin60^{\circ}=\frac{\sqrt{3}}{2}).

Step4: Substitute the values into the formula

(\sin225^{\circ}\cos105^{\circ}=\frac{1}{2}\left(-\frac{1}{2}+\frac{\sqrt{3}}{2}\right)=\frac{-1 + \sqrt{3}}{4}).

Answer:

(\frac{\sqrt{3}-1}{4})