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Chapter 6: Problem 36

Find the inclination \(\theta\) (in radians and degrees) of the line. $$x-\sqrt{3} y+1=0$$

Short Answer

Expert verified
\(\theta =\arctan\left(\frac{1}{\sqrt{3}}\right)\) in radians and \(\theta =\arctan\left(\frac{1}{\sqrt{3}}\right)\cdot \frac{180}{\pi}\) in degrees.

Step by step solution

01

Transform the line equation to the standard form

We rewrite the equation \(x-\sqrt{3}y+1=0\) as \(y=\frac{1}{\sqrt{3}}x-\frac{1}{\sqrt{3}}\). So, the slope of the line \(m\) is equal to \(\frac{1}{\sqrt{3}}\).
02

Calculate inclination in radians

The angle \(\theta\) can be found using the formula \(\tan(\theta) = m\). Applying the inverse tangent function, we get \(\theta=\arctan(m)=\arctan\left(\frac{1}{\sqrt{3}}\right)\).
03

Convert inclination to degrees

To convert radians to degrees, we use the formula \(\theta(\text{in degrees}) = \theta(\text{in radian}) \cdot \frac{180}{\pi}\). Substituting the radian value into this, we get the requirement in 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.

In Exercises \(91-116\), convert the polar equation to rectangular form. $$r=\frac{5}{1-4 \cos \theta}$$

Explain the process of sketching a plane curve given by parametric equations. What is meant by the orientation of the curve?

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

Determine whether the statement is true or false. Justify your answer. If the vertex and focus of a parabola are on a horizontal line, then the directrix of the parabola is vertical.
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