If the measured angular separation between the second minimum to the left of the central maximum and the third minimum to the right of the central maximum is \(30^{\circ}\) in a single slit diffraction pattern recorded using 628 nm light, then the width of the slit is _______ \(\mu \mathrm{m}\).

\(\begin{aligned} & \theta_1=\sin ^{-1}\left(\frac{2 \lambda}{a}\right) \\ & \theta_2=\sin ^{-1}\left(\frac{3 \lambda}{a}\right) \\ & \because \quad \theta_1+\theta_2=30^{\circ}\end{aligned}\)
\(\begin{aligned} & \Rightarrow \sin ^{-1}\left(\frac{2 \lambda}{\mathrm{a}}\right)+\sin ^{-1}\left(\frac{3 \lambda}{\mathrm{a}}\right)=\frac{\pi}{6} \\ & \Rightarrow \frac{2 \lambda}{\mathrm{a}} \sqrt{1-\left(\frac{3 \lambda}{\mathrm{a}}\right)^2}+\frac{3 \lambda}{\mathrm{a}} \sqrt{1+\left(\frac{2 \lambda}{\mathrm{a}}\right)^2}=\sin \frac{\pi}{6}\end{aligned}\)
Here \(\lambda=628 \mathrm{~nm}\)
After solving
\(\mathrm{A}=6.07 \mu \mathrm{~m}\)
Approximate Method :
\(\begin{aligned} & \theta=\theta_1+\theta_2 \\ & \Rightarrow \frac{\pi}{6}=\frac{2 \lambda}{a}+\frac{3 \lambda}{a} \\ & \Rightarrow \frac{\pi}{6}=\frac{5}{a}(628 \mathrm{~nm}) \\ & \Rightarrow a=6 \mu \mathrm{~m}\end{aligned}\)