It is possible to find the exact value of \(\displaystyle{\cos{{\frac{{\pi}}{{{4}}}}}}\) by constructing a right triangle with one angle set to \(\displaystyle{\frac{{\pi}}{{{4}}}}\) radians.

\(\displaystyle{\frac{{\pi}}{{{4}}}}{r}{a}{d}={\frac{{\pi}}{{{4}}}}{r}{a}{d}\times{\frac{{{180}°}}{{\pi}}}\times{r}{a}{d}^{{-{1}}}={45}°\)

Now draw a right triangle with one of the acute angles set to 45 degrees. These two angles would be supplementary, and their sum would be 90 degrees. Thus, the other angle in this triangle would also be 45 degrees, making an isosceles right triangle.

Therefore, \(\displaystyle{\cos{{\frac{{\pi}}{{{4}}}}}}={\cos{{\left({45}\right)}}}={\frac{{{a}{d}{j}.}}{{{h}{y}{p}.}}}={\frac{{{1}}}{{\sqrt{{{2}}}}}}={\frac{{\sqrt{{{2}}}}}{{{2}}}}\)