Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

A guy wire is attached to the top of a radio antenna and to a point on horizontal ground that is \(40.0\) meters from the base of the antenna. If the wire makes an angle of \(58^{\circ} 20^{\prime}\) with the ground, approximate the length of the wire.

Step-by-Step Solution

Answer

The length of the wire is approximately 75.52 meters.

Step 1: Understand the Problem

You're given a right-angled triangle where the wire is the hypotenuse. The distance from the base of the antenna to the point on the ground where the wire is attached is the adjacent side, given as 40 meters. The angle between the wire and the ground is \(58^{\circ} 20^{\prime}\). You need to find the length of the hypotenuse, which is the wire.

Step 2: Convert Angle to Decimal

Convert \(58^{\circ} 20^{\prime}\) to a decimal for easier calculation. One degree is 60 minutes, so \(20\) minutes is \(20/60 = 0.3333\ldots\) degrees. Thus, \(58^{\circ} 20^{\prime}\) is approximately \(58.3333^{\circ}\).

Step 3: Apply the Cosine Rule

Use the cosine function to find the length of the wire (hypotenuse). The cosine of the angle is the adjacent side divided by the hypotenuse: \(\cos(58.3333^{\circ}) = \frac{40}{\text{hypotenuse}}\).

Step 4: Solve for the Hypotenuse

Rearrange the equation to solve for the hypotenuse: \(\text{hypotenuse} = \frac{40}{\cos(58.3333^{\circ})}\). Calculate \(\cos(58.3333^{\circ})\), then divide 40 by this value to find the length of the wire.

Step 5: Calculate the Length

The cosine of \(58.3333^{\circ}\) is approximately 0.5299. Therefore, \(\text{hypotenuse} = \frac{40}{0.5299} \approx 75.52\) meters. So, the length of the wire is approximately 75.52 meters.

Key Concepts

Understanding a Right-Angled Triangle

A right-angled triangle is a type of triangle that features one angle exactly equal to 90 degrees. This unique property makes right-angled triangles very useful in trigonometry. They have three sides called the hypotenuse, adjacent, and opposite. The hypotenuse is the side opposite the right angle and is always the longest side. The other two sides are called the adjacent and opposite sides, and their names depend on the angle you're considering.
In our exercise, the right-angled triangle is formed by the guy wire, the radio antenna, and the ground. Our task involves finding the hypotenuse, which is the length of the wire attached at an angle of 58 degrees and 20 minutes to the ground.

The Cosine Rule Explained

The cosine rule is a crucial component in determining unknown lengths or angles in a triangle, particularly useful in right-angled triangles. In a right-angled triangle, the cosine of one of the non-right angles is defined as the ratio of the length of the adjacent side to the hypotenuse.
In mathematical terms, \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \), where \( \theta \) is the angle in question.
In our scenario, we are given the length of the adjacent side (40 meters) and the angle (58 degrees and 20 minutes). We can rearrange the formula to solve for the hypotenuse: \( \text{hypotenuse} = \frac{\text{adjacent}}{\cos(\theta)} \). This will allow us to find the length of the wire.

Angle Conversion Simplified

Trigonometric calculations often require angles in decimal degrees rather than degrees and minutes. This conversion simplifies calculations and integrates seamlessly with trigonometric functions on calculators. One full degree contains 60 minutes, so to convert minutes into a decimal, divide by 60.
For example, convert \( 58^{\circ} 20^{\prime} \) to decimal: 20 minutes is \( \frac{20}{60} = 0.3333... \). Thus, \( 58^{\circ} 20^{\prime} \) converts to approximately \( 58.3333^{\circ} \).
This conversion is crucial when using calculators to ensure precise results, like when calculating the cosine value for further trigonometric computations.

Hypotenuse Calculation Technique

To calculate the hypotenuse in our right-angled triangle, we apply the understanding of trigonometry through the cosine function. We previously derived that \( \text{hypotenuse} = \frac{\text{adjacent}}{\cos(\theta)} \). Substitute the given angle and side length into the formula for calculation.
First, find the cosine of the decimal-converted angle: \( \cos(58.3333^{\circ}) \approx 0.5299 \).
Now, substitute into our rearranged cosine rule: \( \text{hypotenuse} = \frac{40}{0.5299} \). The result is approximately 75.52 meters.
This final calculation gives us the length of the wire by dividing the length of the adjacent side by the cosine of the angle.