Concept Check Work each problem.What angle does the line y = √3x make with the positive x-axis?

Verified step by step guidance

1. Identify the slope of the line from the equation y = \(\sqrt{3}\)x. The slope (m) is \(\sqrt{3}\).
2. Recall that the tangent of the angle (\(\theta\)) that a line makes with the positive x-axis is equal to the slope of the line. Therefore, \(\tan\)(\(\theta\)) = \(\sqrt{3}\).
3. To find the angle \(\theta\), take the arctangent (inverse tangent) of the slope: \(\theta\) = \(\tan\)^{-1}(\(\sqrt{3}\)).
4. Recognize that \(\tan\)^{-1}(\(\sqrt{3}\)) corresponds to a special angle in trigonometry. Recall the special angles and their tangent values.
5. Determine the angle \(\theta\) in degrees or radians that satisfies \(\tan\)(\(\theta\)) = \(\sqrt{3}\).

Key Concepts

Slope of a Line

The slope of a line in the Cartesian plane is a measure of its steepness, calculated as the ratio of the rise (change in y) to the run (change in x). For the line given by the equation y = √3x, the slope is √3. This slope is crucial for determining the angle the line makes with the positive x-axis.

Tangent of an Angle

In trigonometry, the tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. The slope of a line can be interpreted as the tangent of the angle it makes with the x-axis. Therefore, to find the angle θ that the line y = √3x makes with the x-axis, we can use the relationship tan(θ) = slope.

Inverse Tangent Function

The inverse tangent function, denoted as arctan or tan⁻¹, is used to find an angle when the tangent value is known. Given the slope of the line (√3), we can find the angle θ by calculating θ = arctan(√3). This function is essential for converting the slope back into an angle measurement, which is necessary for answering the original question.