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Chapter 6: Problem 40

Find the angle \(\theta\) (in radians and degrees) between the lines. \(x-y=0\) \(3 x-2 y=-1\)

Short Answer

Expert verified
The angle \(\theta\) between the lines is \(\arctan\frac{1}{5}\) radians or \(\arctan\frac{1}{5} * \frac{180}{\pi}\) degrees.

Step by step solution

01

Find the slopes

To find the slope of a line from its equation, convert the equation to the form \(y = mx + c\) where \(m\) is the slope. For the first line \(x-y=0\), solving for \(y\) gives \(y = x\), so the slope \(m_1=1\). For the second line \(3x - 2y = -1\), solving for \(y\) gives \(y = \frac{3}{2}x + \frac{1}{2}\), so the slope \(m_2 = \frac{3}{2}\).
02

Find the angle in radians

We can calculate the angle \(\theta\) between the two lines using the formula \(\tan{\theta}=\frac{m_2 - m_1}{1 + m_1 m_2}\). Substituting our values, we get \(\tan{\theta}=\frac{\frac{3}{2}-1}{1+1*\frac{3}{2}}=\frac{1}{5}\). Hence, \(\theta = \arctan{\frac{1}{5}}\).
03

Convert to degrees

To convert from radians to degrees, multiply by \(\frac{180}{\pi}\). So, \(\theta = \arctan{\frac{1}{5}}*\frac{180}{\pi}\) degrees.

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Most popular questions from this chapter

Identify the conic as a circle or an ellipse. Then find the center, radius, vertices, foci, and eccentricity of the conic (if applicable), and sketch its graph. \(\frac{x^{2}}{4 / 9}+\frac{(y+1)^{2}}{4 / 9}=1\)

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