Question

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

Explanation:

Step1: Identify the trigonometric ratio

In right - triangle \( \triangle RSQ\) with \( \angle R = 90^{\circ}\), we know the side \(RS = 35\) and we want to find the hypotenuse \(x\) (side \(SQ\)). The trigonometric ratio involving the opposite side (\(RS\)) and the hypotenuse (\(SQ\)) for angle \(Q\) is the sine ratio. The formula for sine is \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, \(\theta = 40^{\circ}\), the opposite side \(a = 35\), and the hypotenuse \(c=x\). So, \(\sin(40^{\circ})=\frac{35}{x}\).

Step2: Solve for \(x\)

Rearrange the formula \(\sin(40^{\circ})=\frac{35}{x}\) to get \(x=\frac{35}{\sin(40^{\circ})}\). We know that \(\sin(40^{\circ})\approx0.6428\). Then \(x=\frac{35}{0.6428}\approx54.4\).

Answer:

\(x\approx54.4\)