/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 43 Find the angle \(\theta\) (in ra... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 43

Find the angle \(\theta\) (in radians and degrees) between the lines. \(x+2 y=8\) \(x-2 y=2\)

Short Answer

Expert verified
The angle \(\theta\) between the two lines is approximately \(-\frac{\pi}{2}\) radians or -90 degrees.

Step by step solution

01

Find the slope for both lines

The slope of the line can be found from the equation in slope-intercept form \(y=mx+b\), where m is the slope. For the first line, rearranging gives \(y = -\frac{1}{2}x + 4\). Hence the slope \(m1 = -\frac{1}{2}\). For the second line, rearranging gives \(y = \frac{1}{2}x + 1\). Hence the slope \(m2 = \frac{1}{2}\).
02

Substitute the slopes to the formula

Substitute \(m1\) and \(m2\) into the formula \[\theta = \arctan\left(\frac{m1-m2}{1+m1 \cdot m2}\right)\] This gives \[\theta = \arctan\left(\frac{-\frac{1}{2} - \frac{1}{2}}{1 + (-\frac{1}{2})\times\frac{1}{2}}\right) = \arctan(-\infty)\]
03

Simplify to get the angle in radians

\[\theta = \arctan(-\infty) \approx -\frac{\pi}{2}\] Note that if the result is negative, it means that the angle is measured in the opposite direction.
04

Convert the angle to degrees

\[\theta = -\frac{180}{\pi} \times \frac{\pi}{2} = -90 ^\circ\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

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. \(9 x^{2}+25 y^{2}-36 x-50 y+60=0\)

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Foci: (卤5,0)\(;\) major axis of length 14

Sketch the graph of the ellipse, using latera recta. \(\frac{x^{2}}{4}+\frac{y^{2}}{1}=1\)

The equations of a parabola and a tangent line to the parabola are given. Use a graphing utility to graph both equations in the same viewing window. Determine the coordinates of the point of tangency. \(y^{2}-8 x=0 \quad x-y+2=0\)

Determine whether the statement is true or false. Justify your answer. It is easier to distinguish the graph of an ellipse from the graph of a circle if the eccentricity of the ellipse is large (close to 1).
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.