Determining the Obtuse Angle Between Two Lines with Given Slopes: A Comprehensive Guide

When dealing with the geometry of lines, one of the key aspects we often need to determine is the angle between two lines, especially the obtuse angle. This article will guide you through the process, using the slopes of the lines. Specifically, we will focus on the lines with slopes of 7 and -3. Let's dive into the detailed steps to find the obtuse angle between these two lines.

Understanding the Slope and the Tangent Function

In the context of lines, the slope (m) represents the rate of change of the y-coordinate with respect to the x-coordinate. Mathematically, the slope is given by the tangent of the angle (θ) that the line makes with the positive x-axis. Therefore, if one of the lines has a slope of 7, it can be expressed as:

tan(θ1) 7

Similarly, for the second line with a slope of -3:

tan(θ2) -3

Calculating the Angles

Using a calculator or a software that can compute the inverse tangent function (arctan), we can find the corresponding angles:

θ1 arctan(7) ≈ 81.87°

θ2 arctan(-3) ≈ -71.56°

Note that the angle returned by arctan for the second line is negative, which means it is measured in the clockwise direction. To convert this to a positive angle in the proper quadrant, we add 180°:

θ2 (positive) -71.56° 180° 108.44°

Calculating the Obtuse Angle

To find the obtuse angle between the two lines, we need to calculate the difference between the absolute values of these angles:

θ |θ1 - θ2| |81.87° - 108.44°| 26.57°

However, the obtuse angle is the larger angle between the two lines, which means we need to subtract this small angle from 180°:

θ_obtuse 180° - 26.57° 153.43°

Understanding the Sign of the Slope

The signs of the slopes play a crucial role in determining the correct quadrant of the angle. The tangent function is positive in the first and third quadrants and negative in the second and fourth quadrants. Here are the key points to remember:

The slope of 7 (θ1) is in the first quadrant, so the angle is 81.87°. The slope of -3 (θ2) is in the fourth quadrant, and its angle is -71.56°, which we convert to a positive angle of 108.44° by adding 180°. The obtuse angle between the two lines is 153.43°.

Conclusion

Determining the obtuse angle between two lines is a fundamental concept in geometry and essential for many practical applications, such as in computer graphics, engineering, and design. Understanding the relationship between the slopes and the angles formed by the lines can help in solving complex problems effectively. By following the steps outlined in this article, you can easily calculate the obtuse angle between two lines with given slopes.

References

If you need to further explore the concepts of slope and tangent, consider consulting textbooks on geometry or online resources that provide detailed explanations and examples. Websites like Mathway, Khan Academy, and WolframAlpha can also serve as valuable tools in understanding and practicing these calculations.