Comparing Angles: Lines with Slopes 2 and 1/2 on X-Axis

Two straight lines have slopes of$2,\frac{1}{2}$.

Which of the lines forms a larger angle with the x axis?

To solve for which line forms a larger angle with the x-axis, we'll proceed by calculating the arctangent of each slope:

• First, calculate the angle for the line with slope $m_1 = 2$:

$\theta_1 = \tan^{-1}(2)$

• Next, calculate the angle for the line with slope $m_2 = \frac{1}{2}$:

$\theta_2 = \tan^{-1}\left(\frac{1}{2}\right)$

Comparing these two results:

• The function $\tan^{-1}(x)$ is increasing, meaning as the value of $x$ increases, so does the angle $\theta$. Therefore, since $2 > \frac{1}{2}$, it follows that $\theta_1 > \theta_2$.

Therefore, the straight line with a slope of 2 forms the largest angle with the x-axis.