Concept Check Work each problem. Find the equation of the line that passes through the origin and makes a 30° angle with the x-axis.

Verified step by step guidance

1. Recognize that the line passes through the origin, so its equation will be of the form \(y = mx\), where \(m\) is the slope of the line.
2. Recall that the slope \(m\) of a line making an angle \(\theta\) with the positive x-axis is given by \(m = \tan(\theta)\).
3. Substitute the given angle \(\theta = 30^\circ\) into the slope formula: \(m = \tan(30^\circ)\).
4. Use the known exact value or expression for \(\tan(30^\circ)\) to find the slope \(m\) (you do not need to calculate the decimal value, just express it).
5. Write the final equation of the line using the slope found: \(y = m x\), where \(m = \tan(30^\circ)\).

Key Concepts

Slope of a Line from an Angle

The slope of a line can be found using the tangent of the angle it makes with the positive x-axis. Specifically, slope m = tan(θ), where θ is the angle between the line and the x-axis. This relationship connects trigonometry with coordinate geometry.

Equation of a Line Through the Origin

A line passing through the origin (0,0) can be expressed as y = mx, where m is the slope. Since the line goes through the origin, there is no y-intercept term, simplifying the equation to a direct proportionality between y and x.

Trigonometric Functions and Angle Measurement

Understanding how to use trigonometric functions like tangent requires knowing angle measurement in degrees or radians. Here, the angle is given as 30°, so converting or directly using tan(30°) helps find the slope accurately.