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Chapter 6: Problem 51

Find the angle \(\theta\) (in radians and degrees) between the lines. $$\begin{aligned} &x+2 y=8\\\ &x-2 y=2 \end{aligned}$$

Short Answer

Expert verified
The angle between the two lines is \(\pi/4\) radians or 45 degrees.

Step by step solution

01

Put the Equations in Slope-Intercept Form

To find the slopes, we first express each given equation in the slope-intercept form, y = mx + b. The values of m will be the slopes of the lines. For the first equation, it will be as follows: \(x + 2y = 8\) becomes \(y = -1/2x + 4\). Secondly, \(x - 2y = 2\) becomes \(y = 1/2x - 1\). Therefore, the slope of the first line (m1) is -1/2 and the slope for the second line (m2) is 1/2.
02

Find the Angle Between the Two Lines

The formula to find the angle between two lines is \(tan(\theta) = \frac{m2 - m1}{1 + m1*m2}\). Substituting the slope values gives us \(tan(\theta) = \frac{1/2 - -1/2}{1 + -1/2*1/2}\). This simplifies to \(tan(\theta) = 1\). Therefore, \(\theta = arctan(1)\), which gives us \(\theta = \pi/4\) radians.
03

Convert the Angle from Radians to Degrees

Now that we have the angle \(\theta\) in radians, we can convert it to degrees by multiplying with 180/蟺. This gives \(\theta = \pi/4 * 180/\pi = 45\) degrees.

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Most popular questions from this chapter

In Exercises \(71-90,\) convert the rectangular equation to polar form. Assume \(a > 0\). $$x^{2}+y^{2}=9 a^{2}$$

In Exercises \(91-116\), convert the polar equation to rectangular form. $$r=\frac{6}{2 \cos \theta-3 \sin \theta}$$

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

In Exercises \(117-126\), convert the polar equation to rectangular form. Then sketch its graph. $$r=2 \sin \theta$$

In Exercises \(91-116\), convert the polar equation to rectangular form. $$\theta=5 \pi / 3$$
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