Question

You slide a box up a loading ramp that is \(3.7 \mathrm{~m}\) long. At the top of the ramp the box has risen a height of \(1.1 \mathrm{~m}\). What is the angle of the ramp above the horizontal?

Short answer

The angle of the ramp is approximately 17.3 degrees.

Step-by-step solution

Understand the Problem

We need to find the angle above the horizontal of a ramp, which forms a right triangle with the horizontal base as the adjacent side, the ramp length as the hypotenuse (3.7 m), and the vertical rise as the opposite side (1.1 m).

Identify the Right Triangle Components

In the context of the problem, the ramp forms the hypotenuse of a right triangle, the height of 1.1 m is the opposite side, and we need to find the angle between the horizontal and the hypotenuse.

Recall and Apply Trigonometric Definitions

To find the angle, we use the sine function, which in a right triangle is defined as the ratio of the length of the opposite side to the hypotenuse: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1.1}{3.7} \).

Solve for the Angle

Calculate \( \theta \) using the inverse sine function: \( \theta = \sin^{-1} \left( \frac{1.1}{3.7} \right) \). Compute the value to find \( \theta \).

Calculate the Numerical Solution

Using a calculator to find the inverse sine, we compute \( \theta \approx \sin^{-1}(0.297) \). This gives \( \theta \approx 17.3^\circ \).

Key concepts

Right Triangle

A right triangle is a special type of triangle in geometry. It has three sides and one of its angles is exactly 90 degrees, known as the right angle. Let's dive deeper into the components of a right triangle:

• The side opposite the right angle is called the hypotenuse. It is always the longest side.
• The other two sides are known as the legs. In our problem, these are the horizontal base of the ramp and the vertical rise.
• The right angle is crucial, as it allows us to use trigonometric concepts to find unknown sides or angles.

In our exercise, the ramp forms a right triangle, where the hypotenuse is the length of the ramp at 3.7 meters. The vertical rise, 1.1 meters in this case, is the opposite leg. The angle we are interested in is between the ramp and the horizontal base. Understanding this setup is the key to applying the correct trigonometric functions.

Sine Function

The sine function is a fundamental concept in trigonometry. It relates the angles of a right triangle to the ratio of specific sides within the triangle. For an angle \( \theta \), the sine function is defined as:

• \( \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \)

This means that if you know the lengths of the opposite side and the hypotenuse, you can calculate the sine of the angle \( \theta \).
In our scenario, the sine function is crucial for calculating the angle of the ramp. We've already identified the opposite side (1.1 m) and the hypotenuse (3.7 m). Therefore, we can express this as:
\( \sin(\theta) = \frac{1.1}{3.7} \).
This ratio serves as a stepping stone to finding the angle using the inverse sine function.

Inverse Sine

The inverse sine function, often written as \( \sin^{-1} \) or arcsin, does one main thing: it helps find the angle when the sine value is known. This function tells us "what angle gives this sine value?"
For example, if \( \sin(\theta) = 0.297 \), then \( \theta = \sin^{-1}(0.297) \). This inverse operation is essential for solving trigonometric problems when angles need to be calculated.

• By using a calculator set in degree mode, you compute \( \theta = \sin^{-1} \left( \frac{1.1}{3.7} \right) \).
• This calculation returns an angle \( \theta \), approximately 17.3 degrees.

This value represents the angle the ramp makes with the horizontal, giving a clear solution to the exercise. Understanding how to use the inverse sine function bridges the gap between knowing side ratios and determining precise angle measures.