Calculating the Acute Angle Between Two Lines with Given Slopes: -√3 and √3

In the field of geometry, determining the angle between two lines can be a common task. This article will delve into a specific instance: finding the acute angle between two lines with slopes -√3 and √3.

1. The Basic Formula for Finding the Angle Between Two Lines

The formula for finding the tangent of the angle θ between two lines with slopes (m_1) and (m_2) is as follows:

tan(θ) (frac{m_1 - m_2}{1 m_1 m_2})

2. Determining the Acute Angle

Given the slopes:

(m_1 -sqrt{3}) (m_2 sqrt{3})

Substitute these values into the formula:

tan(θ) (frac{-sqrt{3} - sqrt{3}}{1 (-sqrt{3})(sqrt{3})})

3. Simplifying the Expression

First, simplify the numerator:

-(sqrt{3}) - (sqrt{3}) -2(sqrt{3})

Next, simplify the denominator:

1 (-(sqrt{3}))((sqrt{3})) 1 - 3 -2

Now, substitute these values back into the formula:

tan(θ) (frac{-2sqrt{3}}{-2}) (sqrt{3})

4. Finding the Angle from the Tangent Value

To find the angle θ that corresponds to the tangent value of (sqrt{3}), we use the inverse tangent function:

θ arctan((sqrt{3})) 60° or (frac{pi}{3}) radians

5. Additional Insights

It is important to note that the tangent of 60° is (sqrt{3}), and the tangent of -60° is also -(sqrt{3}). When considering the acute angle between the lines, we are always interested in the smaller angle, which is 60° or (frac{pi}{3}) radians.

6. Interpretation of Slopes

A slope of -(sqrt{3}) represents an angle of -60°, while a slope of (sqrt{3}) represents an angle of 60°. The acute angle between these lines is the smallest angle formed, which is the difference between the angles represented by each slope, thus 60°.

7. Application with Complex Numbers

To further illustrate, we can use complex numbers. The arguments of the complex numbers -1 i(sqrt{3}) and 1 i(sqrt{3}) are used to find the angle:

Let θ be the acute angle between the lines:

tan(θ) (frac{1 (sqrt{3})}{1 - (sqrt{3})}) or (frac{(sqrt{3}) - (-(sqrt{3}))}{1 - 3} (sqrt{3})

Using the arctangent function: