Question

solve for x. round to the nearest tenth, if necessary.

Explanation:

Step1: Identify the trigonometric ratio

In a right - triangle, we use the cosine function. The cosine of an angle in a right - triangle is defined as \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 18^{\circ}\), the adjacent side to the \(18^{\circ}\) angle is \(EF = 52\), and the hypotenuse is \(DF=x\). So, \(\cos(18^{\circ})=\frac{52}{x}\).

Step2: Solve for \(x\)

Rearrange the equation \(\cos(18^{\circ})=\frac{52}{x}\) to get \(x=\frac{52}{\cos(18^{\circ})}\).
We know that \(\cos(18^{\circ})\approx0.9511\). Then \(x=\frac{52}{0.9511}\approx54.7\).

Answer:

\(54.7\)