Question

solve for x in the triangle. round your answer to the nearest tenth.

Explanation:

Step1: Identify trigonometric ratio

The triangle is right - angled, and we know the hypotenuse (14) and we want to find the side opposite the \(57^{\circ}\) angle (x). The sine function is defined as \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\). So, \(\sin(57^{\circ})=\frac{x}{14}\).

Step2: Solve for x

Multiply both sides of the equation \(\sin(57^{\circ})=\frac{x}{14}\) by 14 to isolate x. We get \(x = 14\times\sin(57^{\circ})\). Using a calculator, \(\sin(57^{\circ})\approx0.8387\). Then \(x=14\times0.8387 = 11.7418\). Rounding to the nearest tenth, we look at the hundredth place. The digit in the hundredth place is 4, which is less than 5, so we round down. So \(x\approx11.7\).

Answer:

\(x\approx11.7\)