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Chapter 6: Problem 52

Find the angle \(\theta\) (in radians and degrees) between the lines. $$\begin{aligned} &3 x-5 y=3\\\ &3 x+5 y=12 \end{aligned}$$

Short Answer

Expert verified
The angle between the two lines is -1.11 radians or -63.43 degrees.

Step by step solution

01

Find the slopes

Rewrite the two equations in the form \(y = mx + c\), where m indicates the slope of each function. The first equation \(-3x+5y=3\) becomes \(y=\frac{3}{5}x + \frac{3}{5}\), so the slope \(m1=\frac{3}{5}\). The second equation \(3x+5y=12\) becomes \(y = -\frac{3}{5}x + \frac{12}{5}\), so the slope \(m2 = -\frac{3}{5}\).
02

Calculate the angle

Use the slopes of the lines in the formula for \(\theta = \arctan \left( \frac{m2-m1}{1+m1 \cdot m2} \right)\). Substituting in the values, we get \(\theta = \arctan \left( \frac{-\frac{3}{5}-\frac{3}{5}}{1+ \frac{3}{5} \cdot -\frac{3}{5}} \right)\). This simplifies to \(\theta = \arctan(-2)\). Taking the arctan of -2 in radians gives us \(-1.11\) radians.
03

Convert radians to degrees

We have the angle in radians but we also need it in degrees. The formula for converting radians to degrees is \(degrees = radians \times \frac{180}{\pi}\). Substituting our radian value into that equation gives us \(-63.43\) degrees.

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Most popular questions from this chapter

In Exercises \(129-132,\) determine whether the statement is true or false. Justify your answer. If \(\theta_{1}=\theta_{2}+2 \pi n\) for some integer \(n,\) then \(\left(r, \theta_{1}\right)\) and \(\left(r, \theta_{2}\right)\) represent the same point in the polar coordinate system.

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