Question

what is the value of x? 4 12 4\sqrt{3} 12\sqrt{2}

Explanation:

Step1: Identify triangle type and trigonometric ratio

This is a right - triangle with one angle \(60^{\circ}\) and hypotenuse \(h = 8\). We want to find the length of the side opposite the \(60^{\circ}\) angle (\(x\)). The sine function is defined as \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). For \(\theta = 60^{\circ}\), \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\).

Step2: Apply the sine formula

We know that \(\sin60^{\circ}=\frac{x}{8}\). Substituting \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\) into the formula, we get \(\frac{\sqrt{3}}{2}=\frac{x}{8}\).

Step3: Solve for \(x\)

To solve for \(x\), we can cross - multiply. So \(x = 8\times\frac{\sqrt{3}}{2}\). Simplifying \(8\times\frac{\sqrt{3}}{2}\), we divide 8 by 2 first, \(8\div2 = 4\), then multiply by \(\sqrt{3}\), so \(x = 4\sqrt{3}\).

Answer:

\(4\sqrt{3}\)