Rearranging the Equation x √3y: Finding the Angle of Inclination and Its Application

In the realm of coordinate geometry, understanding the angle of inclination of a line is crucial for various applications. This article elucidates how to find the angle of inclination for the line equation x √3y, and provides insights into the application of this concept using the 30-60-90 triangle principle.

Understanding the Line Equation

The given equation of the line is x √3y. To find the angle of inclination, it is essential to rearrange this equation into the slope-intercept form, y mx b, where m represents the slope of the line.

Step 1: Rearrangement to Slope-Intercept Form

Let's begin by rearranging x √3y to express y in terms of x:

[ x √3y Rightarrow y frac{1}{√3}x ]

From this rearrangement, we see that the slope, m, of the line is ( frac{1}{√3} ).

Step 2: Calculating the Angle of Inclination

The angle of inclination, denoted as θ, can be calculated using the slope formula:

[ tan(θ) m ]

Substituting the slope, we get:

[ tan(θ) frac{1}{√3} ]

To find θ, we take the arctangent:

[ θ tan^{-1}left(frac{1}{√3}right) ]

The angle ( tan^{-1}left(frac{1}{√3}right) ) corresponds to 30 degrees or ( frac{π}{6} ) radians.

Therefore, the angle of inclination of the line x √3y is:

[ θ 30^circ ]

Application of the 30-60-90 Triangle Principle

When you encounter a slope of ( frac{1}{√3} ) in a line equation, it is helpful to recognize it as the tangent of 30 degrees. This relationship can be visualized through the 30-60-90 triangle, where the sides are in the ratio 1:√3:2. In this triangle, the angle opposite the side of length 1 is 30 degrees.

Rearrangement in Terms of y

Let's rearrange the equation in terms of y to further solidify the understanding:

[ y frac{1}{√3}x ]

Here, y is the vertical change (rise), and x is the horizontal change (run). The ratio ( frac{1}{√3} ) signifies that for every unit increase in x, y increases by ( frac{1}{√3} ).

Angle of Inclination in Degrees

Using the slope formula, we directly obtain:

[ tan(θ) frac{1}{√3} Rightarrow θ 30^circ ]

This confirms that the line y frac{1}{√3}x has an inclination of 30 degrees with respect to the x-axis.

Redefining the Equation for Clarity

For clarity and convenience, we can rewrite the equation as:

[ y frac{x}{√3} ]

It is evident that this line passes through the origin and has a slope of ( frac{1}{√3} ).

Since the slope is given by tan(θ), we have: