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 equation of the given line is \( y = 3x \), which is in the slope-intercept form \( y = mx + b \). Here, the slope \( m \) is 3.

Step 2: Use the tangent function for angle calculation

The angle \( \theta \) between the line and the x-axis can be found using the formula \( \tan\theta = m \), where \( m \) is the slope. In this case, \( \tan\theta = 3 \).

Step 3: Calculate the angle using the arctangent

To find \( \theta \), use the arctangent function: \( \theta = \tan^{-1}(3) \). Use a calculator to find the value of \( \tan^{-1}(3) \), which is approximately \( 71.57^{\circ} \).

Step 4: Confirm the angle is in the first quadrant

Since the angle \( 71.57^{\circ} \) is less than \( 90^{\circ} \), it is indeed a first quadrant angle.

Key Concepts

Understanding Slope

In trigonometry and geometry, the concept of slope is a crucial element in understanding the orientation of a line relative to the x-axis. In its simplest terms, the slope is a measure of how steep a line is. Consider the mathematical form of the slope, represented by the variable \( m \), in the equation of a line \( y = mx + b \). The slope \( m \) dictates how much \( y \) would change with a one-unit increase in \( x \), known as the "rise over run" concept.
For example:

• If \( m = 3 \), as in this problem, it implies that for every unit increase in the horizontal direction (run), the vertical change (rise) is 3 units.
• A positive slope indicates that the line rises as it moves from left to right, while a negative slope indicates a line that falls as it goes from left to right.

Understanding slope is foundational when determining the angle a line makes with the x-axis.

Exploring Arctangent

The arctangent function is a part of the family of inverse trigonometric functions, crucial for solving problems related to angles and slopes. When you encounter a slope \( m \) and need to find the angle it forms with the x-axis, the arctangent function \( \tan^{-1}(x) \) becomes useful. This function finds the angle whose tangent equals the given slope.
To give an example, if \( m = 3 \):

• The formula becomes \( \theta = \tan^{-1}(3) \).
• Using a calculator, you determine that \( \tan^{-1}(3) \approx 71.57^\circ \).

This angle is the angle that the line makes with the x-axis, providing a direct application of the arctangent in finding angles in trigonometry.

Recognizing First Quadrant Angles

In the coordinate plane, a first quadrant angle is an angle that lies between 0 and 90 degrees. Understanding quadrant angles is essential when dealing with directions and angles in geometry. A crucial step in many trigonometric problems is confirming that the angle calculated aligns with expected angle positions.
When determining if an angle is in the first quadrant, consider the following:

• An angle is said to be in the first quadrant if it is positive and less than 90 degrees.
• Since the first quadrant represents the domain where both \( x \) and \( y \) values are positive, angles less than 90° often adhere to this requirement.
• For instance, the angle \( 71.57^\circ \) found in this problem indeed lies in the first quadrant.

Recognizing first quadrant angles helps in ensuring correct interpretation and representation of angles in various mathematical contexts.