Problem 52 - Precalculus Mathematics for Calculus

Question

A 600-ft guy wire is attached to the top of a communications tower. If the wire makes an angle of \(65^{\circ}\) with the ground, how tall is the communications tower?

Step-by-Step Solution

Answer

The communications tower is approximately 543.78 feet tall.

Step 1: Identify the Trigonometric Function

We need to find the height of the communications tower, which can be thought of as the opposite side in a right triangle formed by the guy wire. The guy wire is the hypotenuse, and the angle given is between the wire and the ground. The sine function relates the opposite side to the hypotenuse in a right triangle.

Step 2: Write the Sine Formula

Use the sine function: \[ \sin(65^{\circ}) = \frac{\text{opposite side (height)}}{\text{hypotenuse}} \]Substitute the known value of the hypotenuse (600 ft): \[ \sin(65^{\circ}) = \frac{h}{600} \] where \( h \) is the height of the tower.

Step 3: Solve for the Height

To find the height, rearrange the formula to solve for \( h \):\[ h = 600 \times \sin(65^{\circ}) \] Use a calculator to find \( \sin(65^{\circ}) \):\[ \sin(65^{\circ}) \approx 0.9063 \] Thus, the height \( h \) is:\[ h = 600 \times 0.9063 \approx 543.78 \]

Step 4: Conclusion

The communications tower is approximately 543.78 feet tall.

Key Concepts

Right Triangle

When working with trigonometry, understanding the concept of a right triangle is crucial. A right triangle is a type of triangle that includes a 90-degree angle. It has three sides - the hypotenuse, which is the longest side, and two other sides known as the opposite and adjacent sides.

• The hypotenuse is always opposite the right angle.
• The opposite side is the side opposite the angle of interest (other than the right angle).
• The adjacent side is the side next to the angle of interest and is not the hypotenuse.

In problems like the one with the guy wire and the communication tower, the hypotenuse is the wire itself, forming part of a right triangle with the ground. By understanding these basic components, you can start to see how trigonometric functions are applied.

Sine Function

The sine function is a fundamental component in trigonometry, especially when working with right triangles. Sine of an angle in a right triangle is defined as the ratio between the length of the side opposite that angle and the hypotenuse.

That is, for an angle \(\theta\) in a right triangle, the sine function is expressed as:
\[\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}\]In the context of our problem with the guy wire:

• The angle \(65^{\circ}\) is the angle of interest formed between the wire and the level ground.
• The hypotenuse is the length of the guy wire, which is 600 feet.
• The opposite side is the height of the tower, which we need to determine.

The sine function helps us find this unknown length using the known angle and hypotenuse, signifying its powerful utility in solving right triangle problems.

Angle Measurement

Angle measurement is a pivotal aspect in the study and application of trigonometry. Angles are generally measured in degrees or radians, with one complete revolution around a point equating to 360 degrees or \(2\pi\) radians.

When working with trigonometric problems in right triangles, using angles to determine unknown sides is a common technique. For example, an angle of \(65^{\circ}\) was used in this problem to find the height of the communications tower using trigonometric ratios.

• The angle specified tells us what part of the right triangle's ratio to focus on – here, using sine.
• An angle’s measurement affects which trigonometric function to apply, as each function relates different sides of a triangle.

Working with angles precisely allows us to apply trigonometric functions accurately, solving for unknowns in geometric structures in real-world scenarios.