We would like to find the exact value of \(\displaystyle{\sin{{\frac{{{3}\pi}}{{{4}}}}}}\), First, we can find the reference angle of \(\displaystyle{\frac{{{3}\pi}}{{{4}}}}\) where \(\displaystyle\pi-{\frac{{{3}\pi}}{{{4}}}}={\frac{{\pi}}{{{4}}}}\), so the reference angle is \(\displaystyle{\frac{{\pi}}{{{4}}}}\).

Now the next step is to define the sign of \(\displaystyle{\sin{{\frac{{{3}\pi}}{{{4}}}}}}\). We know that \(\displaystyle{\frac{{{3}\pi}}{{{4}}}}\) is in quadrant 2 which the sine function is positive in this quadrant, so the value of \(\displaystyle{\sin{{\frac{{{3}\pi}}{{{4}}}}}}\) is positive and we can simplify it as follows:

\(\displaystyle\therefore{\sin{{\frac{{{3}\pi}}{{{4}}}}}}={\sin{{\left(\pi-{\frac{{\pi}}{{{4}}}}\right)}}}={\sin{{\frac{{\pi}}{{{4}}}}}}\)

Note that the first step was to find the reference angle and then was to define the sign of \(\displaystyle{\sin{{\frac{{{3}\pi}}{{{4}}}}}}\).

Now we can find the value of \(\displaystyle{\sin{{\frac{{{3}\pi}}{{{4}}}}}}\) by knowing the value of \(\displaystyle{\sin{{\frac{{\pi}}{{{4}}}}}}\) which equals \(\displaystyle{\frac{{\sqrt{{{2}}}}}{{{2}}}}\)

\(\displaystyle\therefore{\sin{{\frac{{{3}\pi}}{{{4}}}}}}={\sin{{\left(\pi-{\frac{{\pi}}{{{4}}}}\right)}}}={\sin{{\frac{{\pi}}{{{4}}}}}}={\frac{{\sqrt{{{2}}}}}{{{2}}}}\)

Now the next step is to define the sign of \(\displaystyle{\sin{{\frac{{{3}\pi}}{{{4}}}}}}\). We know that \(\displaystyle{\frac{{{3}\pi}}{{{4}}}}\) is in quadrant 2 which the sine function is positive in this quadrant, so the value of \(\displaystyle{\sin{{\frac{{{3}\pi}}{{{4}}}}}}\) is positive and we can simplify it as follows:

\(\displaystyle\therefore{\sin{{\frac{{{3}\pi}}{{{4}}}}}}={\sin{{\left(\pi-{\frac{{\pi}}{{{4}}}}\right)}}}={\sin{{\frac{{\pi}}{{{4}}}}}}\)

Note that the first step was to find the reference angle and then was to define the sign of \(\displaystyle{\sin{{\frac{{{3}\pi}}{{{4}}}}}}\).

Now we can find the value of \(\displaystyle{\sin{{\frac{{{3}\pi}}{{{4}}}}}}\) by knowing the value of \(\displaystyle{\sin{{\frac{{\pi}}{{{4}}}}}}\) which equals \(\displaystyle{\frac{{\sqrt{{{2}}}}}{{{2}}}}\)

\(\displaystyle\therefore{\sin{{\frac{{{3}\pi}}{{{4}}}}}}={\sin{{\left(\pi-{\frac{{\pi}}{{{4}}}}\right)}}}={\sin{{\frac{{\pi}}{{{4}}}}}}={\frac{{\sqrt{{{2}}}}}{{{2}}}}\)