Question

a wire connects the top of a 128 - foot tall cell tower to the ground. if the angle between the wire and the ground is 76°, how long is the wire? ? feet round your answer to the nearest hundredth.

Explanation:

Step1: Identify the trigonometric relationship

We have a right - triangle where the height of the cell tower is the opposite side to the given angle and the wire is the hypotenuse. We use the sine function, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Let the length of the wire be $x$. The height of the tower (opposite side) is 128 feet and the angle $\theta = 76^{\circ}$. So, $\sin(76^{\circ})=\frac{128}{x}$.

Step2: Solve for $x$

We can re - arrange the formula $\sin(76^{\circ})=\frac{128}{x}$ to $x=\frac{128}{\sin(76^{\circ})}$. We know that $\sin(76^{\circ})\approx0.9703$. Then $x=\frac{128}{0.9703}\approx131.92$.

Answer:

131.92