Chapter 9: Problem 29

MATHEMATICAL CONNECTIONS Write an expression that can be used to find the measure of the acute angle formed by each line and the x-axis. Then approximate the angle measure to the nearest tenth of a degree. a. \(y=3 x\) b. \(y=\frac{4}{3} x=4\)

### Short Answer

## Step by step solution

## Understanding the concept

## Finding the angle for the first line

## Finding the angle for the second line

## Key Concepts

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

###### Inverse Tangent

The inverse tangent helps us find the acute angle from the slope by essentially 'undoing' the tangent operation. When you use this inverse function on a slope, you get the angle in radians or degrees, depending on your calculator settings. Knowing how to apply this function can be incredibly valuable for students who need to bridge their algebraic understanding with trigonometric concepts.

- Understand that \( \tan^{-1}(m) \) is the angle whose tangent is \( m \).
- Make sure your calculator is set to the correct mode (degree or radian) that your problem requires.
- Use the calculator to compute \( \tan^{-1}(m) \), which gives you the angle measure.

###### Angle Measure

The acute angles formed with the x-axis are particularly interesting. These angles are important because they provide insights into how steep a line is compared to the x-axis.

- An acute angle is always less than 90 degrees.
- In the given problem, we measured angles to the nearest tenth of a degree, ensuring precision.
- The tools in trigonometry help translate between different forms of angle representation, enhancing both visualization and practical application skills.

###### Slope of a Line

The slope describes how steep a line is and whether it ascends, descends, or stays constant. It's a direct measure of a line's inclination:

- A slope of zero indicates a perfectly horizontal line.
- A positive slope means the line ascends as you move from left to right.
- A negative slope implies the line descends as you move from left to right.

###### Acute Angle

Understanding acute angles is essential for correctly applying trigonometric functions, such as inverse tangent, to find angle measures. This is important for solving real-world problems that require an understanding of angle relationships.

- Acute angles are often involved in right triangles, where they complement the 90-degree angle.
- In this exercise, acute angles result from the application of the inverse tangent to slopes derived from linear equations.
- By ensuring that the angle you calculate is an acute angle, you ensure the solution is consistent with the typical geometric context of lines and axes.

###### X-axis

In this context, the x-axis becomes the reference line for determining angles formed by other lines. Any line that doesn't run parallel to the x-axis naturally forms an angle with it. This angle can be calculated by considering the line's slope and using trigonometric functions like inverse tangent.

- Lines intersect the x-axis at y=0.
- The slope tells how sharply a line rises or falls relative to the x-axis.
- Knowing how to find and interpret the angle with the x-axis helps in understanding the geometry of line equations in a coordinate plane.