The Acute Angle Between the Lines 2x^3y5 and 5xy6: A Comprehensive Guide
Introduction
The concept of the acute angle between two lines is a fundamental topic in mathematics, particularly in algebra and geometry. This article will guide you through the step-by-step process of calculating the acute angle between the lines given by the equations 2x^3y5 and 5xy6. We will use basic mathematical concepts to find the slopes (or gradients) of these lines, and then apply a formula to find the acute angle between them.
Step-by-Step Calculation
To find the acute angle between the lines 2x^3y5 and 5xy6, we need to follow a straightforward process involving the following steps:
Convert the given equations into the slope-intercept form (ymx b):
Calculate the gradients (slopes) of the lines:
Use the formula for the tangent of the acute angle between two lines to find the angle:
Step 1: Convert the Equations into the Slope-Intercept Form
Let's start with the first line, 2x^3y5:
2x^3y5
y 5 - 2x/3
This equation is now in the slope-intercept form (ymx b), where the gradient (m) is -2/3.
Now, let's consider the second line, 5xy6:
5xy6
y 6 - 5x
This equation is also in the slope-intercept form, where the gradient (m) is -5.
Step 2: Calculate the Gradients (Slopes) of the Lines
The gradients of the lines are:
For the first line, the gradient is -2/3. For the second line, the gradient is -5.Step 3: Use the Formula for the Tangent of the Acute Angle Between Two Lines
The formula for the tangent of the acute angle between two lines with gradients (m_1) and (m_2) is:
(tan(theta) left| frac{m_2 - m_1}{1 m_1m_2} right|)
Substituting the values of the gradients:
(tan(theta) left| frac{-5 - (-2/3)}{1 (-2/3)(-5)} right|)
(tan(theta) left| frac{-5 2/3}{1 10/3} right|)
(tan(theta) left| frac{-15/3 2/3}{3/3 10/3} right|)
(tan(theta) left| frac{-13/3}{13/3} right|)
(tan(theta) left| -1 right|)
(tan(theta) 1)
Since (tan(theta) 1), the angle (theta) is 45 degrees.
Conclusion
The acute angle between the lines 2x^3y5 and 5xy6 is 45 degrees. This method can be applied to find the angle between any two lines given by their equations in the slope-intercept form. Whether you are a student, a teacher, or a professional in mathematics, mastering this concept is essential for solving more complex problems in algebra and geometry.