Question

find the exact value of the expression.
\\( \sin \left(25^{\circ}\
ight) \cos \left(5^{\circ}\
ight)+\cos \left(25^{\circ}\
ight) \sin \left(5^{\circ}\
ight) \\)
\\( \sin \left(25^{\circ}\
ight) \cos \left(5^{\circ}\
ight)+\cos \left(25^{\circ}\
ight) \sin \left(5^{\circ}\
ight)= \square \\)
(type an exact answer, using radicals as needed. use integers or frac

Explanation:

Step1: Apply the sine addition formula

The formula for \(\sin(A + B)=\sin A\cos B+\cos A\sin B\). Here \(A = 25^{\circ}\) and \(B=5^{\circ}\), so \(\sin(25^{\circ})\cos(5^{\circ})+\cos(25^{\circ})\sin(5^{\circ})=\sin(25^{\circ}+ 5^{\circ})\).

Step2: Simplify the angle

\(25^{\circ}+5^{\circ}=30^{\circ}\), so the expression becomes \(\sin(30^{\circ})\).

Step3: Find the value of \(\sin(30^{\circ})\)

We know that \(\sin(30^{\circ})=\frac{1}{2}\).

Answer:

\(\frac{1}{2}\)