The equation $x^2-3 x y+2 y^2+3 x-5 y+2=0$ represents a pair of straight lines. If $\theta$ is the angle…

The angle between the pair of straight lines represented by $x^2-3 x y+2 y^2+3 x-5 y+2=0$ is determined using the standard formula
for pairs of lines in the general form $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$.

Comparing coefficients, we find $a = 1$, $2h = -3 \implies h = -3/2$, and $b = 2$.
The condition $\Delta = abc + 2fgh - af^2 - bg^2 - ch^2 = 0$ confirms the equation represents a pair of lines.

The angle $\theta$ between the lines is given by $\tan \theta = \left| \frac{2\sqrt{h^2-ab}}{a+b} \right|$.

Substituting values: $h^2 = 9/4$, $ab = 2$, so $h^2 - ab = 1/4$ and $a+b = 3$,
yielding $\tan \theta = \frac{1}{3}$.

Using the identity $\sec^2 \theta = 1 + \tan^2 \theta$, we compute $\sec^2 \theta = 10/9$,
so $\sec \theta = \sqrt{10}/3$, and therefore $\cos \theta = \frac{3}{\sqrt{10}}$.

$\boxed{\frac{3}{\sqrt{10}}}$