$$
Wsintheta=lambda
$$
yielding

$$
sintheta = frac{lambda}{W}.
$$
Here we see that by increasing $lambda$ we also increase $sintheta$. Since the sine is an increasing function of the angle (if it falls in range of $0-90^circ$) we know that the angle here will also increase.

$$
begin{align*}
m &pm 1 \
lambda &= 690 text{ nm} \
theta &pm 23 text{textdegree}
end{align*}
$$

The slit width, $W$ , which produces a first-order diffraction minima can be obtain by applying the relation :

$$
begin{align*}
W sin theta &= m lambda \
W &= dfrac{m lambda}{sin theta} \
&= dfrac{1 (690 text{ nm})}{sin 23 text{textdegree}}
end{align*}
$$

$$
{boxed{W = 1765.92 text{ nm}}}
$$