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Chapter 6: Problem 49

Find the angle \(\theta\) (in radians and degrees) between the lines. $$\begin{array}{r} x-2 y=7 \\ 6 x+2 y=5 \end{array}$$

Short Answer

Expert verified
The angle \(\theta\) between the lines is 1.4 radians or 80.2 degrees

Step by step solution

01

Convert equations into slope-intercept form

Convert the given equations into the slope-intercept form \(y = mx + c\). For the first equation, \(x - 2y = 7\), it can be rewritten as \(y = 0.5x - 3.5\). For the second equation, \(6x + 2y = 5\), it can be rewritten as \(y = -3x + 2.5\)
02

Find the slopes of the lines

For the first equation, the slope \(m1\) is 0.5 and for the second equation, the slope \(m2\) is -3
03

Use the formula to find the angle in radians

Substitute \(m1\) and \(m2\) into the formula mentioned above to get the angle \(\theta\) in radians. This becomes \(\theta = atan(\frac{-3 - 0.5}{1 + (-3)(0.5)})\), which gives \(\theta = atan(-3.5/-2.5) = atan(1.4)\).
04

Convert the angle from radians to degrees

The answer from step 3 is in radians. We convert it to degrees by multiplying by \(\frac{180}{\pi}\). This becomes \(\theta = 1.4 * (180/\pi) = 80.2\) degrees

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Most popular questions from this chapter

In Exercises \(71-90,\) convert the rectangular equation to polar form. Assume \(a > 0\). $$x^{2}+y^{2}-2 a x=0$$

In Exercises \(91-116\), convert the polar equation to rectangular form. $$r=-2 \cos \theta$$

In Exercises \(117-126\), convert the polar equation to rectangular form. Then sketch its graph. $$\theta=3 \pi / 4$$

In Exercises \(117-126\), convert the polar equation to rectangular form. Then sketch its graph. $$r=-3 \sin \theta$$

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