To find the angle between two lines, we use the formula for the tangent of the angle between the lines. If the slopes of the two lines are $m_1$ and $m_2$, then the tangent of the angle $\theta$ between the lines is given by:

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

Step 1: Find the slopes of the lines

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

1. First line: $2x + 3y + 5 = 0$

   • Slope $m_1 = -\frac{2}{3}$

2. Second line: $x + 3y - 2 = 0$

   • Slope $m_2 = -\frac{1}{3}$

Step 2: Substitute into the formula

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

Substitute $m_1 = -\frac{2}{3}$ and $m_2 = -\frac{1}{3}$:

$\tan\theta = \left| \frac{-\frac{1}{3} - \left(-\frac{2}{3}\right)}{1 + \left(-\frac{2}{3}\right)\left(-\frac{1}{3}\right)} \right|$

Simplify the numerator and denominator: $\tan\theta = \left| \frac{-\frac{1}{3} + \frac{2}{3}}{1 + \frac{2}{9}} \right|$

$\tan\theta = \left| \frac{\frac{1}{3}}{\frac{9}{9} + \frac{2}{9}} \right|$

$\tan\theta = \left| \frac{\frac{1}{3}}{\frac{11}{9}} \right|$

$\tan\theta = \left| \frac{1}{3} \times \frac{9}{11} \right|$

$\tan\theta = \frac{3}{11}$

Step 3: Find the angle

The angle $\theta$ is given by: $\theta = \arctan\left(\frac{3}{11}\right)$

Using a calculator or approximation: $\theta \approx 15.26^\circ$

Final Answer:

The angle between the two lines is approximately 15.26 degrees.

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Would you like additional clarification or a step-by-step explanation? Here are related questions to expand your understanding:

1. How do we derive the slope-intercept form from a general equation of a line?
2. How would the calculation change if the lines were parallel?
3. How do we interpret the angle if it’s obtuse instead of acute?
4. What happens to the formula if $m_1 \times m_2 = -1$ (perpendicular lines)?
5. How can we calculate the angle between lines in 3D space?

Tip: Always ensure to simplify fractions completely when working with tangent and slopes to minimize errors.