Problem 55 - Basic Technical Mathematics with Calculus

Question

Solve the given problems.Find the first quadrant angle between the line \(y=3 x\) and the \(x\) -axis.

Step-by-Step Solution

Answer

The angle is approximately \(71.57^\circ\).

Step 1: Identify the slope of the line

The line equation is given as \(y = 3x\). This is in the slope-intercept form \(y = mx + c\), where \(m\) is the slope of the line. Here, \(m = 3\).

Step 2: Use the slope to find the angle

The angle \(\theta\) that the line makes with the positive direction of the x-axis can be determined using the tangent function, \( \tan(\theta) = m\). So, \( \tan(\theta) = 3\).

Step 3: Solve for the angle

To find \(\theta\), we calculate the arctangent of 3. Thus, \( \theta = \arctan(3) \). Use a calculator to find this angle value, which is approximately \(\theta \approx 71.57^\circ\).

Step 4: Ensure the angle is in the first quadrant

Since \(\theta \approx 71.57^\circ\) is already between \(0^\circ\) and \(90^\circ\), it confirms that it is in the first quadrant.

Key Concepts

Slope-intercept form

The slope-intercept form is a common way to express a linear equation. Its general formula is \(y = mx + c\).

• Here, \(y\) and \(x\) are variables that represent any point on the line.
• The letter \(m\) denotes the slope of the line, which shows how steep the line is.
• The term \(c\) is the y-intercept, which is where the line crosses the y-axis.

When you know the slope \(m\) and y-intercept \(c\), you can easily plot the line. In the original exercise, the equation is simplified to \(y = 3x\). This means the line passes through the origin, as there is no y-intercept term. The slope \(m = 3\) indicates a relatively steep line, rising 3 units upward for every 1 unit it moves to the right. Understanding this form helps you quickly identify characteristics of the line.

Arctangent function

The arctangent, often written as \(\arctan\), is an inverse trigonometric function used to find an angle when you know the tangent of that angle. In mathematics, it helps to determine the angle in a right triangle, given the ratio of the opposite side to the adjacent side.

• For a line equation, like \(y = 3x\), the slope \(m\) represents this tangent ratio.
• The arctangent function is used to find the angle \(\theta\) that the line makes with the x-axis.

For example, in the solution provided, \(\tan(\theta) = 3\), so \(\theta = \arctan(3)\). When you calculate this using a calculator, you find that \(\theta\) is approximately \(71.57^\circ\). This process is essential when you need to translate a slope into an actual angle measurement.

First quadrant angle

In a coordinate plane, quadrants are the four sections created by the x and y axes intersecting. The first quadrant is the top-right section, where both \(x\) and \(y\) are positive.When finding an angle a line makes with the x-axis, it’s important to determine which quadrant the angle lies in. A first quadrant angle is between \(0^\circ\) and \(90^\circ\).

• This indicates it's the smaller angle a line can make with the positive x-axis.
• The importance lies in correctly interpreting the angle’s sign and size during calculations.

In the solution, the angle \(\theta \approx 71.57^\circ\), confirming it is a first quadrant angle since it is positive and less than \(90^\circ\). Recognizing this can help prevent errors and ensures the interpretation matches the physical situation described by the line.