Question

(a) If the horizontal distance is doubled but the vertical rise remains the same, will the angle \(\theta\) increase, decrease, or stay the same? (b) Calculate \(\theta\) for \(d_{x}=2(1.33 \mathrm{~m})=2.66 \mathrm{~m}\) and \(d_{y}=0.380 \mathrm{~m}\).

Short answer

(a) The angle \( \theta \) decreases. (b) \( \theta \approx 8.13^\circ \).

Step-by-step solution

Understanding the Problem

The problem involves analyzing the impact on the angle \( \theta \) when the horizontal distance \( d_x \) is changed, but the vertical rise \( d_y \) remains constant. In part (a), we identify whether the angle \( \theta \) will increase, decrease, or stay the same if only \( d_x \) is doubled.

Defining the Relationship Between Angle and Distances

The angle \( \theta \) can be found using the tangent function: \( \tan(\theta) = \frac{d_y}{d_x} \). An increase in \( d_x \) with \( d_y \) constant implies a decrease in \( \tan(\theta) \), thus angle \( \theta \) will decrease.

Calculating the Angle With Doubled Horizontal Distance

Given \( d_x = 2.66 \) m and \( d_y = 0.380 \) m, we calculate \( \theta \) as \( \theta = \tan^{-1} \left( \frac{d_y}{d_x} \right) = \tan^{-1} \left( \frac{0.380}{2.66} \right) \).

Performing the Calculation

Calculate the value: \[ \theta = \tan^{-1} \left( \frac{0.380}{2.66} \right) \approx \tan^{-1}(0.142857) \approx 8.13^\circ \].

Key concepts

Angle Calculation

Angle calculation is a critical concept in both mathematics and physics, serving as a foundation for understanding various phenomena. When tackling problems that involve angles, one often uses trigonometric functions to relate different dimensions or aspects of a shape or path.

In the context of this exercise, we focus on how the angle \( \theta \) is influenced by changes in the horizontal and vertical distances. To calculate \( \theta \), we leverage the inverse tangent function, also written as \( \tan^{-1} \) or \( \arctan \). This function is crucial because it helps us find the angle when given the ratio of the opposite side to the adjacent side in a right triangle.

Steps in angle calculation usually include:

• Identifying the lengths of relevant sides in the context of a geometric shape, such as a triangle.
• Calculating the ratio of these sides if not given directly.
• Using the inverse tangent function to determine the angle from this ratio.

In the given exercise, the calculation steps revealed the angle \( \theta = \tan^{-1} \left( \frac{0.380}{2.66} \right) \), resulting in approximately \( 8.13^\circ \). Understanding how to manipulate these calculations allows us to predict how angles will react to changes in dimensions under various scenarios.

Tangent Function

The tangent function is an essential trigonometric function that describes relationships within right-angled triangles. Specifically, it expresses the ratio of the opposite side to the adjacent side relative to a specific angle. In this exercise, we employ the tangent function to analyze how changes in dimensions affect the angle \( \theta \).

When considering a triangle, the formula for the tangent function is \( \tan(\theta) = \frac{d_y}{d_x} \). Here, \( d_y \) is the vertical rise, and \( d_x \) is the horizontal distance. This relationship shows that any increase in \( d_x \) while keeping \( d_y \) constant will cause \( \tan(\theta) \) to decrease, resulting in a smaller angle.

This principle was applied in the exercise to conclude that doubling the horizontal distance while maintaining the same vertical rise leads to a decrease in the angle \( \theta \). The consequent calculation, using the tangent function, confirmed this with a precise angle measurement.

Understanding the tangent function's role in these calculations is pivotal for students as it helps in grasping how angles interrelate with dimensional changes in physics problems.

Horizontal and Vertical Distances

In physics and trigonometry, understanding horizontal and vertical distances is crucial for solving problems involving right triangles, vectors, and motion. These distances often represent the components of motion or the sides of triangles that are foundational in trigonometric calculations.

Horizontal distance, denoted here as \( d_x \), typically runs parallel to the ground. This dimension often affects how steep or gentle a slope or path appears. In comparison, the vertical distance, \( d_y \), measures elevation or height and is perpendicular to the horizontal aspect.

To determine how these distances affect an angle \( \theta \), one needs to recognize how they alter the ratio used in trigonometric functions. In this exercise, doubling \( d_x \) while keeping \( d_y \) steady created a scenario where the ratio \( \frac{d_y}{d_x} \) became smaller, leading to a decrease in \( \theta \).

This approach highlights the importance of manipulating and balancing these distances. By mastering how to adjust and apply both horizontal and vertical measurements, students can enhance their ability to solve complex trigonometric and physics problems efficiently.