Problem 15 - Algebra 2 and Trigonometry

Question

An access ramp reaches a doorway that is 2.5 feet above the ground. If the ramp is 10 feet long, what is the sine of the angle that the ramp makes with the ground?

Step-by-Step Solution

Answer

The sine of the angle that the ramp makes with the ground is 0.25.

Step 1: Understand the Problem

We need to find the sine of the angle that the ramp makes with the ground. Given data: the ramp's length is 10 feet, and the height to the doorway is 2.5 feet.

Step 2: Identify the Relevant Trigonometric Function

We know that sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse. Therefore, \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \). Here, the opposite side is the height of the doorway (2.5 feet), and the hypotenuse is the length of the ramp (10 feet).

Step 3: Calculate the Sine of the Angle

Substitute the known values into the sine formula: \( \sin(\theta) = \frac{2.5}{10} \). Simplify the fraction to get \( \sin(\theta) = 0.25 \).

Step 4: Conclusion

Thus, the sine of the angle that the ramp makes with the ground is 0.25, indicating that the angle created by the ramp is roughly 14.5 degrees using a calculator's inverse sine function.

Key Concepts

Understanding the Sine Function

The sine function is one of the basic trigonometric functions used to relate angles to ratios of the sides of a right triangle.
In a right triangle, if you consider one of the non-right angles, the sine of that angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse.
The formula for sine is:

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

This means that if we know the lengths of these two sides, we can easily calculate the sine of the angle.
In the context of this problem, the hypotenuse is the length of the ramp, while the opposite side is the height the ramp reaches. By plugging these values into our formula, we calculate the sine as 0.25.
Understanding this concept is essential as it provides a practical way to find the angle size without direct measurement.

Exploring Right Triangles

Right triangles are one of the most important shapes in trigonometry. They are defined by having one angle that is exactly 90 degrees, often referred to as the right angle.
The other two angles will always add up to 90 degrees in a right triangle. The longest side in a right triangle, opposite the right angle, is called the hypotenuse.
The other two sides are the opposite and adjacent sides. They are named in relation to the angle you are examining in the triangle.
For this particular problem:

• The hypotenuse is the ramp, at 10 feet.
• The opposite side is the doorway height, at 2.5 feet.

Right triangles are fundamental in the study of trigonometry because they simplify the relationships between angles and side lengths using these trigonometric ratios. By mastering the concept of right triangles, solving geometry problems becomes significantly easier.

Angle Calculation Using Trigonometry

Calculating angles in right triangles can often be simplified using trigonometric functions. By knowing the specific sides' lengths, you can directly compute an angle's measure.
In our example with the ramp, once we know the sine of the angle using the formula \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\), we can easily find the angle using the inverse sine function, often denoted as \(\sin^{-1}\).
For our problem, we found \(\sin(\theta) = 0.25\).

• To find \(\theta\), use the calculator's inverse sine feature: \(\theta = \sin^{-1}(0.25)\).
• This gives us an angle of approximately 14.5 degrees.

Trigonometry provides a way to determine this without measuring equipment, making it an incredibly valuable tool in mathematics. Recognizing the relationships between sides and angles through functions such as sine can offer efficient solutions to real-world problems involving angles.