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Chapter 6: Problem 27

Find the inclination \(\theta\) (in radians and degrees) of the line passing through the points. $$(-\sqrt{3},-1),(0,-2)$$

Short Answer

Expert verified
The inclination of the line is \(\arctan(-1 / \sqrt{3})\) in radians. The same inclination in degrees can be calculated by multiplying the radian measure by 180/\(\pi\).

Step by step solution

01

Calculate the Slope

The slope (m) of a line passing through points \((x1, y1)\) and \((x2, y2)\) is given by the formula \(m = (y2 - y1) / (x2 - x1)\). For the given points (-\(\sqrt{3}\), -1) and (0, -2), the slope will be: \(m = (-2 - (-1)) / (0 - (-\sqrt{3})) = -1 / \sqrt{3}\).
02

Calculate the Inclination in Radians

The inclination of a line is the angle \(\theta\) its makes with the positive x-axis. It is given by the equation \(\theta = \arctan(m)\). Here, \(m = -1 / \sqrt{3}\), thus \(\theta = \arctan(-1 / \sqrt{3})\). This is the inclination in radians.
03

Convert the Radian Measurement to Degrees

Degrees can be obtained from radian measure by multiplying by 180 and dividing by \(\pi\). Hence, after having the radian measure of the inclination, it can be converted to degrees using the given formula.

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Most popular questions from this chapter

In Exercises \(117-126\), convert the polar equation to rectangular form. Then sketch its graph. $$\theta=\pi / 6$$

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

In Exercises \(91-116\), convert the polar equation to rectangular form. $$r=4$$

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.

Find the distance between the point and the line. Point \((1,-3)\) Line \(4 x-3 y=-7\)
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