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Chapter 5: Problem 51

What angle does the line \(y=\sqrt{3} x\) make with the positive \(x\) -axis?

Short Answer

Expert verified
The angle is 60掳 or \frac{\pi}{3} radians.

Step by step solution

01

Understanding the Slope of the Line

The given line equation is in the form of slope-intercept form, which is written as \( y = mx \) where \( m \) is the slope. Identify the slope \( m \) of the given line. Here, \( m = \sqrt{3} \).
02

Using the Tangent Function

Recall that the slope of a line is the tangent of the angle \( \theta \) it makes with the positive x-axis. Therefore, we can write the equation: \( \tan(\theta) = m \). Substitue \( m \) with \( \sqrt{3} \).
03

Solving for \( \theta \)

Solve for \( \theta \) by taking the inverse tangent (arctangent) of \( \sqrt{3} \) using a calculator or referencing trigonometric tables: \( \theta = \arctan(\sqrt{3}) \).
04

Finding the Specific Angle

Calculate \( \arctan(\sqrt{3}) \), which is a known trigonometric value. The angle is \( \theta = 60^\circ \) or \( \theta = \frac{\pi}{3} \text{ radians} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

slope-intercept form
To understand the angle a line makes with the x-axis, we start with its slope-intercept form. The slope-intercept form of a line is given by the equation
\( y = mx + b \), where:
  • \( m \) is the slope of the line
  • \( b \) is the y-intercept
For our particular problem, the equation is
\( y = \sqrt{3} x \).
Here,
  • \( m = \sqrt{3} \) and,
  • the y-intercept \( b \) is 0.
The slope, \( m \), tells us how steep the line is. It indicates the ratio of the change in the y-coordinate to the change in the x-coordinate as you move along the line. In other words, for every unit increase in \( x \), the value of \( y \) increases by \( m \) times. This foundational concept is essential in translating the slope into an angle.
tangent function
After identifying the slope, we relate it to the angle using the tangent function. The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. In the coordinate plane, this translates to the slope of the line:
\( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = m \).
In our problem,
  • the slope \( m = \sqrt{3} \).
Given the equation \( \tan(\theta) = \sqrt{3} \), we use the tangent function to find the angle \( \theta \). It means that the tangent of the angle formed by the line with the positive x-axis is equal to the slope of the line. This relationship helps bridge the algebraic concept of slope with the geometric concept of angles.
inverse tangent
To determine the angle from its tangent, we use the inverse tangent, or arctangent function. If \( \tan(\theta) = m \), the angle \( \theta \) is given by \( \theta = \arctan(m) \).
Using our identified slope
  • \( m = \sqrt{3} \),
we have
\( \theta = \arctan(\sqrt{3}) \).
Using a trigonometric table or calculator, we find
\( \arctan(\sqrt{3}) = 60 ^\circ \) or \( \frac{\pi}{3} \text{ radians} \).
This result tells us that the angle formed by the line with the positive x-axis is 60 degrees. This method of using the inverse tangent is crucial for finding the angle when given the slope.

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Most popular questions from this chapter

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