Problem 78 - Precalculus : Building Concepts and Connections

Question

In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal places. In Milwaukee, Wisconsin, the building code states that for a ramp to qualify as handicapped accessible, it can rise only 1 foot for every 8 feet of horizontal length. What is the degree of incline for the ramp to the nearest thousandth of a degree? (Source: www.mkedcd.org)

Step-by-Step Solution

Answer

The degree of incline for the ramp to the nearest thousandth of a degree after rounding is: \(\Theta = \text{rounded}(\text{atan}(\frac{1}{8}) \times \frac{180}{\pi}, 3)\)

Step 1: Identify Variables

In this problem, the rise of the ramp is 1 foot and the run is 8 feet. So, the ratio of the rise to the run is \(\frac{1}{8}\). This ratio can be used to calculate the angle of incline.

Step 2: Apply Tangent Function

In a right triangle, the tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. The angle we want to find is the one between the hypotenuse (ramp) and the adjacent side (horizontal length). So, for our triangle: \(\tan(\Theta) = \frac{1}{8}\)

Step 3: Evaluate Angle Using Inverse Tangent

To find the angle that corresponds to this tangent value, we use the inverse tangent function (arctangent). The angle \(\Theta\) in degrees can be found by converting the result from radians to degrees (since most calculators return this type of functions in radians): \(\Theta = \text{atan}(\frac{1}{8}) \times \frac{180}{\pi}\)

Step 4: Round Result

The final result should be rounded to three decimal places to give the angle to the nearest thousandth of a degree. The degree of incline is: \(\Theta = \text{rounded}(\text{atan}(\frac{1}{8}) \times \frac{180}{\pi}, 3)\)

Key Concepts

Tangent Function

The tangent function is a key concept in trigonometry, particularly when dealing with right triangles. It's a ratio, comparing the side opposite to a given angle to the side adjacent to that angle in a right triangle. In simpler terms, for a right triangle with angle \( \theta \), the formula for the tangent is \( \tan(\theta) = \frac{\textrm{opposite side}}{\textrm{adjacent side}} \).

This function is one of the main trigonometric ratios and is vitally important when trying to find unknown angles or lengths in right-angled triangles. For students working with inclined planes or ramps, like in the Milwaukee building code example, tangent ratios provide a direct method to determine the angle of incline just by knowing the 'rise' and 'run'.

Inverse Tangent Function

The inverse tangent function, also known as the arctangent, is used to find an angle when the value of the tangent is known. The symbol for the inverse tangent is \( \text{atan} \) or \( \tan^{-1} \). So, when you have a ratio from a right triangle's sides, you can find the measure of the angle using the inverse tangent.

For example, if you know the tangent of an angle is \( \frac{1}{8} \) like in our exercise, you could determine the angle in question by calculating \( \Theta = \text{atan}(\frac{1}{8}) \) using a calculator. This will initially give you an angle in radians, which most commonly needs to be converted into degrees to make sense in everyday scenarios. The conversion is straightforward: just multiply the radians by \( \frac{180}{\pi} \) to get the degrees.

Trigonometric Ratios

Trigonometric ratios are essential tools in trigonometry. These ratios, namely sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. In a right triangle, each of these ratios defines a specific relationship:

• Sine (sin) is the ratio of the opposite side to the hypotenuse.
• Cosine (cos) is the ratio of the adjacent side to the hypotenuse.
• Tangent (tan) is the ratio of the opposite side to the adjacent side.

These relationships remain constant for a given angle, making them incredibly useful for a variety of applications such as physics, engineering, and, as seen in our exercise, architecture and design when compliance with building codes is essential.

Degree of Incline

The degree of incline is a measure of how steep a slope is. In the context of ramps and accessibility, the incline is critical as it affects how easily the ramp can be used, especially for those with mobility challenges. The degree of incline is usually expressed as an angle.

In the context of trigonometry, once the rise over the run ratio is determined, you can calculate the degree of incline of a ramp by finding the angle whose tangent is equal to that ratio. This application demonstrates the practical use of trigonometry in ensuring safe and accessible designs in accordance with regulations like the building code in Milwaukee, Wisconsin.