Understanding the Angle Between a Line with Slope √3 and a Slope of -1

In this article, we explore the angle between a line with a slope of √3 and a line with a slope of -1. We'll dive into the mathematical principles and solve the problem step by step, explaining the trigonometric concepts involved, and discuss the angles obtained.p>

The Concept of Slope

Slope is a fundamental concept in geometry and algebra, representing the inclination or gradient of a line. The slope of a line can be expressed as the tangent of the angle it makes with the positive x-axis. For instance, if a line's slope is √3, it indicates that the angle it makes with the x-axis is 60 degrees.

Using the Tangent Function to Find the Angle

Given two lines with slopes m1 √3 and m2 -1, we aim to find the angle between these two lines. The formula to find the angle between two lines, based on their slopes, is:

tan(θ) (m2 - m1) / (1 m1 * m2)

Let's break down the problem using this formula.

Solving the Problem

First, substitute the given values into the formula:

tan(θ) (-1 - √3) / (1 (-1) * (√3))

Resolve the numerator:

tan(θ) -1 - √3

Resolve the denominator:

1 (-1) * (√3) 1 - √3

Therefore, we have:

tan(θ) (-1 - √3) / (1 - √3)

To simplify this expression, multiply the numerator and denominator by the conjugate of the denominator:

tan(θ) [(-1 - √3) * (1 √3)] / [(1 - √3) * (1 √3)]

Apply the identity (a - b)(a b) a^2 - b^2:

tan(θ) [(-1 - √3)(1 √3)] / [1 - (√3)^2]

Simplify the numerator and denominator:

tan(θ) (-1 - 2√3 - 3) / (1 - 3) (-4 - 2√3) / -2

Further simplify:

tan(θ) 2 √3

However, this is not the correct form to identify the angle. We need to use the basic properties of the tangent function to find the angles.

Using the Tangent Formula Correctly

Let's use the correct formula for the tangent of the angle between two lines:

tan(θ) |(m2 - m1) / (1 m1 * m2)|

Substitute the given values:

tan(θ) |(-1 - √3) / (1 - √3)|

Since we know that tan(60°) √3, we can use the properties of the tangent function:

tan(θ) |(-1 - √3) / (1 - √3)|

This simplifies to:

tan(θ) |(√3 - 1)/(1 √3)|

We know that the angles can be found by using the inverse tangent function:

θ1 arctan(√3 - 1)

θ2 180° - arctan(√3 - 1)

The angles are approximately:

θ1 75°

θ2 105°

Hence, the angles between the pair of lines with slopes √3 and -1 are 75° and 105°.

Conclusion

In conclusion, we have explored the mathematical concepts behind finding the angles between two lines given their slopes. The angles obtained are 75° and 105°. This knowledge is valuable in various fields, including geometry, physics, and engineering, where understanding line angles is crucial.

Keywords:

Angle between lines, slope calculation, tangent function, trigonometry