To find the angle between two lines given by their equations, we can use the following formula:

$\tan \theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|$

where $m_1$ and $m_2$ are the slopes of the two lines.

Step 1: Find the slopes $m_1$ and $m_2$ of the lines.

The equation of a line in the form $ax + by + c = 0$ has a slope $m = -\frac{a}{b}$.

Line 1: $x - 2y + 10 = 0$

• Rewrite as $2y = x + 10$ or $y = \frac{1}{2}x + 5$.
• Thus, the slope $m_1 = \frac{1}{2}$.

Line 2: $3x - y - 4 = 0$

• Rewrite as $y = 3x - 4$.
• Thus, the slope $m_2 = 3$.

Step 2: Substitute $m_1$ and $m_2$ into the formula.

$\tan \theta = \left| \frac{\frac{1}{2} - 3}{1 + \frac{1}{2} \cdot 3} \right| = \left| \frac{\frac{1 - 6}{2}}{1 + \frac{3}{2}} \right| = \left| \frac{-\frac{5}{2}}{\frac{5}{2}} \right| = 1$

Step 3: Find $\theta$ by taking the arctangent.

$\theta = \arctan(1) = 45^\circ$

Therefore, the angle between the two lines is $45^\circ$.

Would you like a more detailed explanation or have any questions?

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Related Questions:

1. How would the calculation change if one of the slopes was zero?
2. Can we find the angle between two lines if their equations are given in a different form, like slope-intercept?
3. How does this method apply to parallel lines?
4. What if the two lines were perpendicular? How would we determine the angle?
5. Can this method be used for finding angles between lines in three-dimensional space?

Tip: When using trigonometric functions, ensure your calculator is set to the correct angle mode (degrees or radians) based on the problem requirements.