Question

find the exact value of the expression
sin \frac { 5 pi } { 4 } cos \frac { pi } { 12 } + cos \frac { 5 pi } { 4 } sin \frac { pi } { 12 }
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Use the sine addition formula

The formula for \(\sin(A + B)=\sin A\cos B+\cos A\sin B\).
Let \(A = \frac{\pi}{12}\) and \(B=\frac{5\pi}{12}\). Then \(\sin\frac{\pi}{12}\cos\frac{5\pi}{12}+\cos\frac{\pi}{12}\sin\frac{5\pi}{12}=\sin(\frac{\pi}{12}+\frac{5\pi}{12})\).

Step2: Simplify the sum of angles

Calculate \(\frac{\pi}{12}+\frac{5\pi}{12}=\frac{\pi + 5\pi}{12}=\frac{6\pi}{12}=\frac{\pi}{2}\).

Step3: Evaluate the sine function

Since \(\sin\frac{\pi}{2}=1\).

Answer:

\(1\)