Question

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

Explanation:

Step1: Identify the trigonometric ratio

In right - triangle \(HFG\), we know the side opposite to the angle \(71^{\circ}\) (\(HG = 23\)) and we need to find the hypotenuse (\(x=HF\)). We use the sine ratio. The sine of an angle \(\theta\) in a right - triangle is defined as \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). So, \(\sin71^{\circ}=\frac{HG}{HF}\).

Step2: Solve for \(x\)

Substitute \(HG = 23\) and \(HF=x\) into the sine formula: \(\sin71^{\circ}=\frac{23}{x}\). Then, we can solve for \(x\) by cross - multiplying: \(x=\frac{23}{\sin71^{\circ}}\).
Since \(\sin71^{\circ}\approx0.9455\), then \(x=\frac{23}{0.9455}\approx24.3\).

Answer:

\(x\approx24.3\)