To solve this problem, we need to find the length of the guy wire that is attached to the top of a 400-foot tall tower, making a $57.6^\circ$ angle with the tower.

Solution

1. Identify the Triangle:

   • This forms a right triangle where:
     • The height of the tower (opposite side) = 400 feet.
     • The angle between the guy wire and the tower = $57.6^\circ$.
     • The hypotenuse (the guy wire) = unknown (let's call it $x$).

2. Set Up the Equation Using Trigonometry:

   • Since we know the opposite side (tower height) and are looking for the hypotenuse, we use the sine function: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
   • Plugging in the values: $\sin(57.6^\circ) = \frac{400}{x}$

3. Solve for $x$: $x = \frac{400}{\sin(57.6^\circ)}$

4. Calculate the Result:

   • First, find $\sin(57.6^\circ)$: $\sin(57.6^\circ) \approx 0.844$
   • Substitute back to find $x$: $x = \frac{400}{0.844} \approx 474.17 \text{ feet}$

Final Answer:

The length of the guy wire is approximately 474.2 feet, rounded to the nearest tenth of a unit.

Would you like further details or have any questions?

Here are some additional questions related to this problem:

1. What trigonometric function would be used if we had the angle and the adjacent side instead?
2. How would you solve for the angle if you were given the length of the guy wire?
3. What changes in the calculation if the angle were with the ground instead of the tower?
4. How can we use the cosine or tangent functions to solve similar problems?
5. What tools can help ensure accuracy in trigonometric calculations?

Tip: Always double-check your angle mode (degrees vs. radians) when performing trigonometric calculations on a calculator.