Question

find the exact value of the expression
sin \frac { pi } { 12 } cos \frac { 3 pi } { 4 } + cos \frac { pi } { 12 } sin \frac { 3 pi } { 4 }
sin \frac { pi } { 12 } cos \frac { 3 pi } { 4 } + cos \frac { pi } { 12 } sin \frac { 3 pi } { 4 } = (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 \(B = \frac{3\pi}{4}\). Then \(\sin\frac{\pi}{12}\cos\frac{3\pi}{4}+\cos\frac{\pi}{12}\sin\frac{3\pi}{4}=\sin(\frac{\pi}{12}+\frac{3\pi}{4})\).

Step2: Simplify the sum of angles

First, find a common denominator for \(\frac{\pi}{12}+\frac{3\pi}{4}\). Since \(\frac{3\pi}{4}=\frac{9\pi}{12}\), then \(\frac{\pi}{12}+\frac{9\pi}{12}=\frac{\pi + 9\pi}{12}=\frac{10\pi}{12}=\frac{5\pi}{6}\).

Step3: Evaluate the sine function

We know that \(\sin\frac{5\pi}{6}\). Since \(\sin\theta\) has a value of \(\frac{1}{2}\) when \(\theta=\frac{\pi}{6}+ 2k\pi\) or \(\theta=\pi-\frac{\pi}{6}+2k\pi,k\in\mathbb{Z}\), and \(\frac{5\pi}{6}=\pi-\frac{\pi}{6}\), so \(\sin\frac{5\pi}{6}=\frac{1}{2}\).

Answer:

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