find the exact value of the expression\n sin \frac { pi } { 12 } cos \frac { 3 pi } { 4 } + cos \frac { pi }…

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})