Introduction

In this article, we will explore the geometric relationship between a line and the x-axis, focusing on determining the angle a line with the equation $sqrt{3}x - y - 5 0$ produces with the x-axis. We will use the slope of the line, represented in the slope-intercept form, to find this angle.

Understanding the Equation and Its Slope

The given equation is $sqrt{3}x - y - 5 0$. To find the angle the line produces with the x-axis, we first need to rewrite the equation in the slope-intercept form, which is expressed as $y mx b$, where $m$ is the slope of the line and $b$ is the y-intercept.

Rewriting the Equation

Starting with the original equation:

$sqrt{3}x - y - 5 0$

We can rearrange it to solve for $y$:

$-y -sqrt{3}x 5$

$y sqrt{3}x - 5$

From this, we see that the slope $m$ of the line is $sqrt{3}$.

Calculating the Angle

The angle $theta$ that the line makes with the positive x-axis can be found using the formula:

$tan theta m$

Substituting the slope into the equation:

$tan theta sqrt{3}$

To find $theta$, we take the arctangent:

$theta tan^{-1}(sqrt{3})$

The value of $tan^{-1}(sqrt{3})$ corresponds to an angle of $60^circ$, which means the line makes a $60^circ$ angle with the positive x-axis.

Interpreting the Angle

However, the angle with the x-axis is typically expressed as a positive angle in the anti-clockwise direction. Therefore, the angle the line produces with the x-axis is:

$theta 180^circ - 60^circ 120^circ$

Thus, the angle that the straight line produces with the x-axis is $boxed{120^circ}$.

Further Analysis and Considerations

The above process can also be applied to the equation $y -sqrt{3}x - 5$, which is in the slope-intercept form $y mx b$ where the slope $m -sqrt{3}$. Using the arctangent function, we find:

$tan theta -sqrt{3}$

The angle $theta$ corresponding to this value is either $-60^circ$ (below the x-axis) or $120^circ$ (above the x-axis).

Conclusion

In conclusion, the angle a line with the equation $sqrt{3}x - y - 5 0$ produces with the x-axis is $120^circ$. This analysis can be used to further understand the geometric relationship between lines and the x-axis in various mathematical contexts, such as in trigonometry and calculus.