Problem 22 - College Trigonometry

Question

Using a right triangle with a horizontal leg of length 100 and vertical leg with length 7 , show that a \(7 \%\) grade means that the road (hypotenuse) makes about a \(4^{\circ}\) angle with the horizontal. (It will not be exactly \(4^{\circ}\), but it's pretty close.)

Step-by-Step Solution

Answer

The angle of the road is approximately 4 degrees.

Step 1: Understand the Problem

We have a right triangle where the horizontal leg (adjacent side) is 100 units long, and the vertical leg (opposite side) is 7 units long. We need to calculate the angle between the hypotenuse and the horizontal leg, which represents the road's incline, and show that it is approximately 4 degrees.

Step 2: Calculate the Tangent of the Angle

The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Here, tangent is given by \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{7}{100} \).

Step 3: Find the Angle Using Inverse Tangent

To find the angle \( \theta \), we take the inverse tangent (arctan) of the ratio we found: \( \theta = \arctan\left(\frac{7}{100}\right) \). Calculating \( \arctan(0.07) \) gives us the angle in degrees.

Step 4: Approximate the Angle

Using a calculator, find \( \theta \approx 4.00^{\circ} \). This confirms that the angle is approximately 4 degrees, matching closely to the given hint.

Key Concepts

The Right Triangle

In trigonometry, one of the first shapes you encounter is a right triangle. This triangle has one angle precisely equal to 90 degrees. Because of this specific angle, right triangles have unique properties that make them incredibly useful in various calculations. When dealing with any right triangle, the longest side is known as the hypotenuse, whereas the other two sides are referred to as the legs of the triangle.
In our example, we have a right triangle with a horizontal leg (adjacent) measuring 100 units and a vertical leg (opposite) measuring 7 units. These dimensions help us apply trigonometric concepts to calculate angles and understand inclines, like that of a road. Understanding how a right triangle works is crucial because it helps us employ trigonometric functions like tangent.
A right triangle makes it possible to apply relationships between angles and sides, providing the foundation for evaluating the geometry of more complex forms.

Understanding Tangent

The tangent is a key trigonometric function that plays a vital role in connecting angles and side ratios in a right triangle. In simple terms, the tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. For a given angle \(\theta\), the tangent is given by the formula:

• \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

In our exercise, the opposite side is 7, and the adjacent side is 100. Therefore, the tangent of the angle \(\theta\) is \(\tan(\theta) = \frac{7}{100} \). This ratio provides a number that trigonometric tables or calculators can use to find the actual angle in degrees. The tangent function is critical in determining angles when the sides of a triangle are known, making it a go-to tool for engineers and architects.

Discovering the Angle with Inverse Tangent

Inverse tangent, often referred to as arctan, is used to find an angle when the tangent ratio is known. If you know the tangent ratio (like \(\frac{7}{100}\) from our exercise), you can determine the angle by applying the inverse tangent function. In terms of formula, this is stated as:

• \( \theta = \arctan(\text{tangent value}) \)

For our triangle, we use \( \theta = \arctan(\frac{7}{100}) \) to find the angle. With a calculator, \( \arctan(0.07) \) gives you approximately \(4.00^{\circ}\).
Using the inverse tangent function is powerful because it connects side ratios to angles without needing complex calculations. This makes it a valuable function in various practical applications, such as assessing road grades, architectural design, and even technology involving satellite communications where angles are crucial.