# Evaluate the integrals. \int_{0}^{\frac{\pi}{2}}0\sqrt{1-\cos 20}d0

Evaluate the integrals.
${\int }_{0}^{\frac{\pi }{2}}0\sqrt{1-\mathrm{cos}20}d0$
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Step 1
We have to evaluate the integral ${\int }_{0}^{\frac{\pi }{2}}\sqrt{1-\mathrm{cos}20}d0$
We know that,
$1-\mathrm{cos}20=2{\mathrm{sin}}^{2}0$
Therefore,
${\int }_{0}^{\frac{\pi }{2}}\sqrt{1-\mathrm{cos}20}d0={\int }_{0}^{\frac{\pi }{2}}\sqrt{2{\mathrm{sin}}^{2}0}d0$
$={\int }_{0}^{\frac{\pi }{2}}\sqrt{2}\cdot \mathrm{sin}0d0$
$=\sqrt{2}{\int }_{0}^{\frac{\pi }{2}}\mathrm{sin}0d0$
$=\sqrt{2}{\left[-\mathrm{cos}0\right]}_{0}^{\frac{\pi }{2}}$
$=\sqrt{2}\left[-\mathrm{cos}\frac{\pi }{2}+\mathrm{cos}0\right]$
$=\sqrt{2}\left[-0+1\right]$
$=\sqrt{2}$
Step 2
Hence, required answer is $\sqrt{2}$.