Understanding the Slope and Angle Between Lines

In mathematics, particularly in coordinate geometry, finding the slope of a line when provided with the angle between two lines and the slope of one of these lines is a fundamental concept. This article will guide you through the process step by step and provide clear examples for each scenario.

Identifying the Given Information

To accurately calculate the slope of a line when given the angle between two lines and the slope of one, start by clearly identifying the following:

The slope of the known line, denoted as (m_1) The angle between the two lines, denoted as (theta) The goal is to find the slope of the other line, denoted as (m_2)

Using the Tangent of the Angle

The relationship between the slopes of two lines and the angle between them is given by the formula:

[tan theta frac{m_2 - m_1}{1 m_1 m_2}]

To find (m_2), rearrange this equation:

[tan theta (1 m_1 m_2) m_2 - m_1]

Further simplification provides:

[tan theta - m_1 m_1 tan theta cdot m_2 m_2 - m_1]

Collect like terms:

[tan theta - m_1 m_1 tan theta cdot m_2 m_2 - m_1]

((1 tan theta cdot m_1) m_2 m_2 tan theta]

(m_2 frac{m_1 (1 - tan theta)}{1 - m_1 tan theta})

Calculating (m_2)

Now that the equation is solved, substitute the values for (m_1) and (theta) to find (m_2). Remember to convert (theta) to radians if necessary.

Example:
(m_1 2), (theta 45^circ)

First, calculate (tan 45^circ 1).

Plug into the formula:

[m_2 frac{2 cdot 1}{1 - 2 cdot 1} frac{2}{1 - 2} frac{2}{-1} -2]

Therefore, the slope (m_2) of the other line is (-2).

Checking the Angle After a Rotational Change

Let's examine a situation where a line rotates anti-clockwise by 45 degrees, and the slope after rotation is given as 3. We need to determine the slope before rotation, denoted as (m).

The relationship is given by:

[tan 45^circ 3 - m] [1 frac{3 - m}{1 cdot 3}] [frac{3 - m}{3} 1] [3 - m 3] [m frac{1}{2} 0.5]

The angle of the required line before rotation is (arctan 0.5 approx 26^circ 34, text{minutes}).

Understanding the Illustration

Consider the following illustration: A line starts from point A and makes an angle (theta_1) with the positive x-axis. It is rotated by an angle (theta) to make an angle (theta_2) with the x-axis. The angle between the lines is (theta theta_2 - theta_1).

The formula for the tangent of the angle between two lines is derived as:

[tan theta frac{tan theta_2 - tan theta_1}{1 tan theta_2 cdot tan theta_1}]

Given (theta 45^circ) and (tan theta 1), and (tan theta_2 3), we need to find (theta_1).

From the equation:

[1 frac{3 - tan theta_1}{1 3 tan theta_1}] [1 3 tan theta_1 3 - tan theta_1] [4 tan theta_1 2] [tan theta_1 frac{1}{2}]

Conclusion

In cases where the slope of the line with respect to the x-axis before rotation is lesser than the slope after rotation, (tan theta_1

References

[Angle between Two Straight Lines, Angle between Two Intersecting Lines]