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Chapter 6: Problem 54

Find the angle \(\theta\) (in radians and degrees) between the lines. $$\begin{aligned} &0.02 x-0.05 y=-0.19\\\ &0.03 x+0.04 y=0.52 \end{aligned}$$

Short Answer

Expert verified
The angle \(\theta\) is \(\arctan\left(-1.5\right)\) radians or \(\arctan\left(-1.5\right) \times \frac{180}{\pi}\) degrees. However, the value depends on arctan -1.5, which must be calculated by a calculator.

Step by step solution

01

Calculation of the slopes

First find the slopes of the given lines. For the first line: \(m_1 = -\frac{a_1}{b_1} = -\frac{0.02}{-0.05} = 0.4\)For the second line: \(m_2 = -\frac{a_2}{b_2} = -\frac{0.03}{0.04} = -0.75\)
02

Calculating the angle in radians

Then, we plug these values into the formula \(\theta = \arctan \left( \frac{m_2 - m_1}{1 + m_1 m_2} \right)\) to get the angle in radians. \(\theta = \arctan\left( \frac{-0.75 - 0.4}{1 + (0.4\times -0.75)} \right) = \arctan\left( -1.5 \right)\)
03

Convert angle to degrees

We know that \(1 \text{ radian} = \frac{180}{\pi} \text{ degrees}\), so multiply the radian measure by this factor to convert to degrees.\(\theta_{\text{degree}} = \theta_{\text{radian}} \times \frac{180}{\pi}= \arctan\left( -1.5 \right) \times \frac{180}{\pi}\)

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Most popular questions from this chapter

True or False? Determine whether the statement is true or false. Justify your answer. The conic represented by the following equation is a parabola. \(r=\frac{6}{3-2 \cos \theta}\)

Determine whether the statement is true or false. Justify your answer. A line that has an inclination greater than \(\pi / 2\) radians has a negative slope.

In Exercises \(91-116\), convert the polar equation to rectangular form. $$r=-5 \sin \theta$$

In Exercises \(117-126\), convert the polar equation to rectangular form. Then sketch its graph. $$r=-6 \cos \theta$$

Explain the process of sketching a plane curve given by parametric equations. What is meant by the orientation of the curve?
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