Question

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

Explanation:

Step1: Identify the trigonometric ratio

In right - triangle \(TYZ\) with \(\angle Z = 20^{\circ}\), hypotenuse \(TZ = 40\), and we want to find the side \(x\) (opposite to \(\angle Z\)). Use the sine ratio: \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\).

Step2: Substitute the values into the sine formula

We know that \(\theta = 20^{\circ}\), opposite \(=x\), and hypotenuse \(= 40\). So, \(\sin(20^{\circ})=\frac{x}{40}\).

Step3: Solve for \(x\)

Multiply both sides of the equation by \(40\): \(x = 40\times\sin(20^{\circ})\).
Since \(\sin(20^{\circ})\approx0.3420\), then \(x = 40\times0.3420\).
\(x=13.68\approx13.7\)

Answer:

\(x = 13.7\)