 # Find the exact value of the trigonometric function at the given real number. NS Ayaana Buck 2021-10-16 Answered
Find the exact value of the trigonometric function at the given real number.
$$\displaystyle{\sin{{\frac{{{3}\pi}}{{{4}}}}}}$$

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