Step 1: Understanding the Question:
The angle between a pair of lines represented by a second-degree equation depends only on its homogeneous part $ax^2 + 2hxy + by^2$.
Step 2: Key Formula or Approach:
1. $\tan \theta = |\frac{2\sqrt{h^2 - ab}}{a+b}|$.
2. If $\tan \theta = x/y$, then $\cos \theta = \frac{y}{\sqrt{x^2 + y^2}}$.
Step 3: Detailed Explanation:
1. Compare $x^2 - 3xy + 2y^2 + ... = 0$ with $ax^2 + 2hxy + by^2 + ... = 0$.
We get $a = 1, b = 2, 2h = -3 \implies h = -3/2$.
2. Calculate $\tan \theta$:
$\tan \theta = |\frac{2\sqrt{(-3/2)^2 - 1(2)}}{1 + 2}| = |\frac{2\sqrt{9/4 - 2}}{3}| = |\frac{2\sqrt{1/4}}{3}| = \frac{2 \cdot (1/2)}{3} = \frac{1}{3}$.
3. Find $\cos \theta$:
Since $\tan \theta = 1/3$, the opposite side is 1 and adjacent is 3.
Hypotenuse $= \sqrt{1^2 + 3^2} = \sqrt{10}$.
$\cos \theta = \frac{3}{\sqrt{10}}$.
Step 4: Final Answer:
The value of $\cos \theta$ is $\frac{3}{\sqrt{10}}$.