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Chapter 6: Problem 38

Find the angle \(\theta\) (in radians and degrees) between the lines. \(x+3 y=2\) \(x-2 y=-3\)

Short Answer

Expert verified
The angle \(\theta\) between the two lines is about 0.59 radians or 33.69 degrees.

Step by step solution

01

Find the slopes

Firstly, need to rewrite both lines in slope-intercept form \(y = mx + n\), where m is the slope. So, rewrite \(x+3y=2\) to \(y=\(-\frac{1}{3}\)x + \frac{2}{3}\) and \(x-2y=-3\) to \(y=\(\frac{1}{2}\)x + \frac{3}{2}\). Here, the slopes for these two lines are \(-\frac{1}{3}\) and \(\frac{1}{2}\) respectively.
02

Use the tangent formula

The angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) can be found using the formula \(\tan (\theta) = \frac{m_2 - m_1}{1 + m_1 m_2}\). Substitute \(m_1=-\frac{1}{3}\) and \(m_2=\frac{1}{2}\) into the formula. So, \(\tan (\theta) = \frac{\frac{1}{2} - -\frac{1}{3})}{1 + -\frac{1}{3} * \frac{1}{2}} = \frac{5}{6}\).
03

Evaluate the arctangent

To find the value for \(\theta\), use the arctangent (or inverse tangent) function: \(\theta = \arctan(\frac{5}{6})\). Calculate this result in radians and degrees.

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