Question

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

Explanation:

Step1: Identify trig - ratio

In right - triangle FGH, we know the hypotenuse \(HG = 23\) and we want to find the adjacent side \(x\) to the angle \(\angle F=71^{\circ}\). We use the cosine function. The cosine of an angle in a right - triangle is defined as \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), so \(\cos(71^{\circ})=\frac{x}{23}\).

Step2: Solve for \(x\)

Multiply both sides of the equation \(\cos(71^{\circ})=\frac{x}{23}\) by 23 to isolate \(x\). We get \(x = 23\times\cos(71^{\circ})\).
Since \(\cos(71^{\circ})\approx0.326\), then \(x = 23\times0.326 = 7.498\).

Step3: Round the result

Rounding \(7.498\) to the nearest tenth gives \(x\approx7.5\).

Answer:

\(7.5\)