Question

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

Explanation:

Step1: Identify the trigonometric ratio

In right - triangle \(MLK\), we know the side \(LK = 8.5\) (opposite to the angle \(64^{\circ}\)) and we need to find the hypotenuse \(MK=x\). The sine ratio is defined as \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). So, \(\sin64^{\circ}=\frac{LK}{MK}\).

Step2: Substitute the values and solve for \(x\)

Substituting \(LK = 8.5\) and \(MK=x\) into the sine formula, we get \(\sin64^{\circ}=\frac{8.5}{x}\). Then, \(x=\frac{8.5}{\sin64^{\circ}}\). Since \(\sin64^{\circ}\approx0.8988\), then \(x=\frac{8.5}{0.8988}\approx9.5\).

Answer:

\(x\approx9.5\)