Question

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

Explanation:

Step1: Identify the trigonometric ratio

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, \(\theta = 35^{\circ}\), the hypotenuse \(TZ = 3.2\), and the side opposite to \(\angle Z\) is \(TY=x\).

Step2: Apply the sine formula

We know that \(\sin Z=\frac{TY}{TZ}\). Substituting the values, \(\sin(35^{\circ})=\frac{x}{3.2}\).

Step3: Solve for \(x\)

Multiply both sides of the equation by \(3.2\): \(x = 3.2\times\sin(35^{\circ})\).
Since \(\sin(35^{\circ})\approx0.5736\), then \(x=3.2\times0.5736 = 1.83552\approx1.8\)

Answer:

\(x\approx1.8\)