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Chapter 4: Problem 55

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

Short Answer

Expert verified
The angle is approximately \( 71.57^{\circ} \).

Step by step solution

01

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.
02

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, \( \tan\theta = 3 \).
03

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 \( 71.57^{\circ} \).
04

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.

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Key Concepts

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

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掳 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.

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Most popular questions from this chapter

Solve the given problems. Sketch an appropriate figure, unless the figure is given. When the distance from Earth to the sun is 94,500,000 mi, the angle that the sun subtends on Earth is \(0.522^{\circ} .\) Find the diameter of the sun.

Use a calculator to verify the given relationships or statements. \(\left[\sin ^{2} \theta=(\sin \theta)^{2}\right]\).$$\frac{\sin 43.7^{\circ}}{\cos 43.7^{\circ}}=\tan 43.7^{\circ}$$

Solve the given problems. Sketch an appropriate figure, unless the figure is given. A rectangular piece of plywood \(4.00 \mathrm{ft}\) by \(8.00 \mathrm{ft}\) is cut from one corner to an opposite corner. What are the angles between edges of the resulting pieces?

Designate each angle by the quadrant in which the terminal side lies, or as a quadrantal angle. $$31^{\circ}, 310^{\circ}$$

Use a calculator to verify the given relationships or statements. \(\left[\sin ^{2} \theta=(\sin \theta)^{2}\right]\).$$\sin 78.4^{\circ}=2\left(\sin 39.2^{\circ}\right)\left(\cos 39.2^{\circ}\right)$$
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