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Chapter 6: Problem 33

Find the inclination \(\theta\) (in radians and degrees) of the line. \(6 x-2 y+8=0\)

Short Answer

Expert verified
The inclination of the line \(6x - 2y + 8 = 0\) is \(\arctan(-3)\) radians or \(\arctan(-3) \times \frac{180}{\pi}\) degrees.

Step by step solution

01

Rewrite the equation in the form \(y=mx + b\)

Start by rearrange the equation \(6x - 2y + 8 = 0\) to isolate y. This can be done by first moving \(6x + 8\) to the right, and then dividing by \(-2\) to isolate \(y\). Doing so gives us the equation in the form \(y = -3x - 4\).
02

Find the slope \(m\)

By comparing the equation \(y = -3x - 4\) to the standard form \(y = mx + b\), we can see that the slope of the line, denoted by \(m\), is \(-3\).
03

Use the slope to find the inclination in radians

Knowing the slope, we can say that \(\tan(\theta) = m\). Therefore the inclination angle \(\theta = \arctan(m)\). Substituting \(m = -3\) gives us \(\theta = \arctan(-3)\). Calculate this value with a calculator to get the result in radians.
04

Convert radians to degrees

To convert the angle from radians to degrees, use the formula \(\theta_{\text{deg}} = \theta_{\text{rad}} \times \frac{180}{\pi}\). Substitute \(\theta_{\text{rad}}\) with the result from step 3 to get \(\theta_{\text{deg}}\) in degrees.

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