Question

\(.\) Logical REASONING A line with a positive slope passes through the origin, making a \(60^{\circ}\) angle with the positive \(x\) -axis. Write an equation of the line.

Short answer

The equation of the line is \(y = \sqrt{3}x\).

Step-by-step solution

Find the slope

The slope \(m\) of the line can be found using the fact that the tangent of the angle the line makes with the x-axis is equal to the slope. The angle in this case is \(60^{\circ}\), hence \(m = \tan(60^{\circ})\).

Calculate the slope

Using the tangent value for \(60^{\circ}\), which is \(\sqrt{3}\), find the slope: \(m = \tan(60^{\circ}) = \sqrt{3}\).

Write the equation of the line

Substitute the calculated slope \(m\) into the slope-intercept form of the line equation \(y = mx + b\). Since the line passes through the origin, \(b\) is 0. Hence the equation of the line is \(y = \sqrt{3}x\).