Problem 67 - 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. A 20 -foot-long piece of wire is attached to the top of a pole at one end and nailed to the ground at the other end. If the wire makes an angle of \(30^{\circ}\) with the ground, find the height of the pole.

Step-by-Step Solution

Answer

Therefore, the height of the pole is 10 feet.

Step 1: Identify the Problem

First, let's visualize the problem. We have a pole, a wire, and the ground forming a right triangle. The wire is the hypotenuse of the triangle, the pole is the opposite side, and the ground is the adjacent side. The angle of 30 degrees is formed by the ground and the wire.

Step 2: Set Up the Equation

Since we need to find the height of the pole (opposite side) and we know the length of the wire (hypotenuse), we can use the sine function. It is defined as sin(angle) = opposite/hypotenuse. By substituting the given values, we get an equation: sin(30) = height / 20.

Step 3: Solve the Equation

Solve the equation for the unknown 'height'. From the table of sine values, we know that sin(30) = 0.5. Substituting this value, we get 0.5 = height / 20. Multiplying both sides by 20 gives us the height of the pole, which is 0.5*20 = 10 feet.

Key Concepts

Trigonometric Functions in Right Triangles

Trigonometric functions allow us to connect angles in a triangle to the ratios of its sides. In right triangle trigonometry, the three main functions are sine (sin), cosine (cos), and tangent (tan). They are particularly useful when trying to find missing side lengths or angles within a right triangle. Each function is based on ratios:

• Sine: \( \sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}} \)
• Cosine: \( \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} \)
• Tangent: \( \tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}} \)

In our exercise, we used the sine function because we had information about the angle and the hypotenuse, and needed to find the opposite side, which is the height of the pole. Using these functions makes solving right triangle problems straightforward by turning geometric problems into simple algebraic equations.

Understanding and Using the Sine Rule

The sine rule is a specific trigonometric property that is particularly useful for solving problems involving non-right triangles, but in a right triangle, we often refer to specific trigonometric identities like sine. The sine function helps relate the angle to the sides of the triangle in right triangles. In our problem, since we wanted to find the height of the pole (opposite side) and we had the hypotenuse and angle, the sine function was perfect. The sine of a 30-degree angle is known to be 0.5, a key fact that helps solve many trigonometric problems easily because 30 degrees is a standard angle with easily memorized sine and cosine values.After identifying the equation \( \sin(30) = \frac{\text{height}}{20} \), solving it was a matter of multiplying the hypotenuse by the sine of the angle, making this a simple multiplication problem. This illustrates the efficiency and utility of the sine function in right triangle problems.

Applying Right Triangle Trigonometry to Real-World Problems

Trigonometry isn't just for the classroom—it has many real-world applications. Every time we encounter situations that involve angles and distances, trigonometry can help us derive useful information. This could include measuring heights of structures where direct measurement is difficult, such as a pole or building, or navigating using angles and distances in fields like architecture, construction, and even astronomy. In such real-world applications, knowing how to decompose these problems into right triangle trigonometry problems can help solve them more efficiently. Start by visualizing the problem and identifying the right triangle and its components. Then, use trigonometric functions to relate the known values to those you want to find. Being able to convert complex real-world phenomena into calculable problems involving angles and side lengths makes trigonometry invaluable in real-world scenarios, just like it did in finding the height of the pole in our problem.