Question

Applications The distance between two stakes on a slope is taped at 2055 ft, and the angle of the slope with the horizontal is \(12.3^{\circ} .\) Find the horizontal distance between the stakes.

Short answer

The horizontal distance is 2055 ft multiplied by the cosine of 12.3 degrees.

Step-by-step solution

Identify Known Values

We know the length of the slope, which is 2055 ft and the angle of the slope with the horizontal, which is 12.3 degrees.

Understand the Relationship

The problem describes a right-angled triangle where the slope distance (2055 ft) is the hypotenuse, and the angle with the horizontal (12.3 degrees) helps us find the horizontal distance, which is the adjacent side of the angle in the triangle.

Use the Cosine Function

The cosine of an angle in a right triangle is equal to the length of the adjacent side divided by the length of the hypotenuse. We can express this as: \(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\). Plugging in the known values gives us \(\cos(12.3^\circ) = \frac{\text{Horizontal Distance}}{2055 \text{ ft}}\).

Solve for the Horizontal Distance

Multiply both sides by the hypotenuse to solve for the horizontal distance: \(\text{Horizontal Distance} = 2055 \text{ ft} \times \cos(12.3^\circ)\).

Calculate the Cosine and the Distance

Use a calculator to find \(\cos(12.3^\circ)\) and then multiply by 2055 ft to find the horizontal distance.

Key concepts

Understanding Trigonometry

Trigonometry is a branch of mathematics that focuses on the relationships between the sides and angles of triangles, particularly right-angled triangles. It is a foundational concept that finds applications in various disciplines, including physics, engineering, navigation, and even art.

When dealing with slope distance calculation, trigonometry comes into play because we are essentially dealing with the dimensions of a triangle formed by the slope, the horizontal ground, and the line connecting them (the hypotenuse). The basic trigonometric functions are sine, cosine, and tangent, each representing a ratio between two sides of a right-angled triangle.

For our specific exercise, we know the hypotenuse (the slope distance) and need to calculate the adjacent side (the horizontal distance). To achieve this, we use the cosine function, as it relates the adjacent side to the hypotenuse with respect to the given angle.

The Cosine Function

The cosine function is one of the three primary trigonometric functions, and it is immensely useful in calculating distances in right-angled triangles. In our context, it defines the ratio of the adjacent side to the hypotenuse when the angle is given. The formula for this relationship is: \[\[\begin{align*}\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}\end{align*}\]\]
For the exercise improvement, it is essential to remember that the cosine function will always yield a result between -1 and 1. This is because it represents the cosine of an angle, which is a projection of a radius onto the x-axis in a unit circle. Consequently, when you multiply the cosine of the angle by the hypotenuse (the slope distance), you will get the actual length of the adjacent side in linear terms, which is the horizontal distance we seek to find.

Navigating Right-Angled Triangles

Right-angled triangles are the cornerstone of trigonometry. They have one angle that is exactly 90 degrees, and this unique feature makes them particularly easy to analyze using trigonometric functions. In the case of our slope distance calculation, envisioning the scenario as a right-angled triangle with one side along the horizontal (the ground), one side vertical (elevation change), and the hypotenuse (the tape measuring the slope) is crucial.

Knowing that we have a right-angled triangle allows us to apply the cosine function accurately. In practice, students should pay attention to correctly identifying the right angle, the adjacent side (the side we want to find, in this case, the horizontal distance), the opposite side, and the hypotenuse from the information given to ensure the correct application of trigonometric functions.