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Chapter 6: Problem 71

Determine whether the statement is true or false. Justify your answer. A line that has an inclination greater than \(\pi / 2\) radians has a negative slope.

Short Answer

Expert verified
The statement is true. A line that has an inclination greater than \(\pi / 2\) radians has a negative slope.

Step by step solution

01

Understand the concept of slope

The slope of a line in a plane is a measure of the steepness of the line, or its 'slant'. Positive slope means that the line is rising to the right, and negative slope means that the line is falling to the right.
02

Understand the concept of inclination in radians

Inclination or angle of the line with the positive direction of x-axis (counter clock-wise) is generally expressed in radians in the interval \([0, 2\pi)\). If the angle is measured counter-clockwise from the positive x-axis, then angles between \(0\) and \(\pi / 2\) radians fall in the first quadrant, angles between \(\pi / 2\) and \(\pi\) radians fall in the second quadrant, angles between \(\pi\) and \(3\pi / 2\) radians fall in the third quadrant, and angles between \(3\pi / 2\) and \(2\pi\) radians fall in the fourth quadrant.
03

Analyze the inclination of the line and its slope

A line that has an inclination greater than \(\pi / 2\) radians and less than \(\pi\) radians falls in the second quadrant. A line in the second quadrant has a negative slope because it falls to the right.
04

Conclusion

Based on the above analysis, we can conclude that the given statement 'A line that has an inclination greater than \(\pi / 2\) radians has a negative slope' is true.

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Most popular questions from this chapter

The path of a softball is modeled by \(-12.5(y-7.125)=(x-6.25)^{2},\) where the coordinates \(x\) and \(y\) are measured in feet, with \(x=0\) corresponding to the position from which the ball was thrown. (a) Use a graphing utility to graph the trajectory of the softball. (b) Use the trace feature of the graphing utility to approximate the highest point and the range of the trajectory.

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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. \(\frac{(x+3)^{2}}{12}+\frac{(y-2)^{2}}{16}=1\)
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