Answer: The value of the expression is [tex]\frac{1+\sqrt{3}} {4}[/tex].
Explanation:
The given expression is,
[tex]\sin (\frac{3\pi}{4})\cos ( \frac{\pi}{12} )[/tex]
Step 1: Break the angles.
[tex]\sin (\pi-\frac{\pi}{4})\cos (\frac{\pi}{4}-\frac{\pi}{3} )[/tex]
Step 2: Use quadrant concept to find the value of [tex]\sin (\pi-\frac{\pi}{4})[/tex]
[tex]\sin (\frac{\pi}{4})\cos (\frac{\pi}{4}-\frac{\pi}{3} )[/tex]
Step 3:Use [tex]\cos (A-B)=\cos A\cos B+\sin A\sin B[/tex]
[tex]\sin (\frac{\pi}{4})[\cos (\frac{\pi}{4})\cos (\frac{\pi}{3})+\sin (\frac{\pi}{4})\sin (\frac{\pi}{3})][/tex]
Step 4: Put these values by using trigonometric table.
[tex](\frac{1}{\sqrt 2})[(\frac{1}{\sqrt 2})(\frac{1}{2})+(\frac{1}{\sqrt 2})(\frac{\sqrt3}{2})][/tex]
[tex](\frac{1}{\sqrt 2})[(\frac{1}{2\sqrt 2})+(\frac{\sqrt3}{2\sqrt 2})][/tex]
[tex](\frac{1}{\sqrt 2})(\frac{1+\sqrt3}{2\sqrt 2})[/tex]
[tex]\frac{1+\sqrt3}{4}[/tex]
Therefore, the value of the expression is [tex]\frac{1+\sqrt{3}} {4}[/tex].
