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Chapter 6: Problem 89

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.

Step by step solution

01

Understanding the inclination angle

Inclination angle of a line is the angle between the positive x-axis and the line when measured in anticlockwise direction. The inclination angle ranges between 0 and \(2\pi\) radians.
02

Relating slope to the inclination angle

The slope of a line is positive if the line is inclined in the first or third quadrant (0 to \(\pi / 2\) radians or \(\pi\) to \(\frac{3\pi}{2}\) radians), while the slope is negative when the line is inclined in the second or fourth quadrant (\(\pi / 2\) to \(\pi\) radians or \(\frac{3\pi}{2}\) to \(2\pi\) radians).
03

Evaluating the statement

A line with an inclination greater than \(\pi / 2\) radians means the line is in the second or fourth quadrant. According to step 2, this makes the slope negative. Therefore the statement is true.

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