As shown in the figure, two coherent sources S_1 and S_2 emit sound...

Let the distance between the two coherent sources $S_1$ and $S_2$ be $d = 3\lambda$.

For a point $P$ on the circular wire of large radius at an angular position $\theta$, the rays from $S_1$ and $S_2$ are approximately parallel.

The path difference $\Delta x$ between the waves reaching point $P$ from $S_1$ and $S_2$ is given by:

$\Delta x = d \cos \theta$

Substituting $d = 3\lambda$, we get:

$\Delta x = 3\lambda \cos \theta$

For constructive interference, the path difference must be an integral multiple of the wavelength $\lambda$:

$\Delta x = n\lambda$

Equating the two expressions for path difference:

$3\lambda \cos \theta = n\lambda$

$\Rightarrow \cos \theta = \dfrac{n}{3}$

Since we need the angular positions in the first quadrant, $\theta$ lies in the range $[0^\circ, 90^\circ]$, which implies $0 \le \cos \theta \le 1$.

The possible integer values for $n$ are $0, 1, 2,$ and $3$.

For $n = 3$, $\cos \theta = 1 \Rightarrow \theta = 0^\circ$

For $n = 2$, $\cos \theta = \dfrac{2}{3} \Rightarrow \theta = \cos^{-1}\left(\dfrac{2}{3}\right)$

For $n = 1$, $\cos \theta = \dfrac{1}{3} \Rightarrow \theta = \cos^{-1}\left(\dfrac{1}{3}\right)$

For $n = 0$, $\cos \theta = 0 \Rightarrow \theta = 90^\circ$

Thus, the angular positions for constructive interference in the first quadrant are $0^\circ, \cos^{-1}\left(\dfrac{2}{3}\right), \cos^{-1}\left(\dfrac{1}{3}\right), 90^\circ$.