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Chapter 6: Problem 32

Find the inclination \(\theta\) (in radians and degrees) of the line passing through the points. $$(0,100),(50,0)$$

Short Answer

Expert verified
The inclination of the line passing through the points is -1.1071 radians or -63.43 degrees. If a positive angle less than 180潞 is needed, then the inclination is 116.57潞.

Step by step solution

01

Calculate Slope of the Line

The slope of the line is calculated by using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where (x_1, y_1) and (x_2, y_2) are the coordinates of the given points. Placing the coordinates of points (0,100) and (50,0) into the formula, we find \(m = \frac{0 - 100}{50 - 0} = -2\).
02

Find the Inclination in Radians

The inclination of a line is the angle that line makes with the positive x-axis, moving counterclockwise. Since the slope of the line is \(m = -2\), the tangent of the inclination angle \(\theta = m\). So, the inclination angle in radians is the inverse tangent of the slope \(m = -2\). Using an arc tangent function, we have \(\theta = atan(-2) = -1.1071 \) radians. We find that the angle is negative, which indicates that the line is inclined in the clockwise direction.
03

Convert Radians to Degrees

To convert radians into degrees, we use the formula \(\text{degrees} = \text{radians} \times \frac{180}{\pi}\). Using the value of \(\theta = -1.1071 \) radians in the formula, we get \( -1.1071 \times \frac{180}{\pi} = -63.43 ^\circ \). This means that the inclination of the line in degrees is -63.43 degrees.
04

Determine the Correct Inclination

In some use cases, the inclination must be a positive value and must be less than 180潞. If so, we simply add 180潞 to the angle we found, obtaining \(180潞 - 63.43潞 = 116.57潞\). If you need the result in radians, convert 116.57潞 back into radians.

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