Question

x =
number
x =
y =
color
y =

Explanation:

Step1: Use the sine function for \(x\)

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here \(\theta = 30^{\circ}\), the hypotenuse \(c = 12\sqrt{3}\), and the opposite side to \(30^{\circ}\) is \(x\). So \(\sin30^{\circ}=\frac{x}{12\sqrt{3}}\). Since \(\sin30^{\circ}=\frac{1}{2}\), we have \(\frac{1}{2}=\frac{x}{12\sqrt{3}}\). Cross - multiply: \(x=\frac{12\sqrt{3}}{2}=6\sqrt{3}\).

Step2: Use the cosine function for \(y\)

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here \(\theta = 30^{\circ}\), the hypotenuse \(c = 12\sqrt{3}\), and the adjacent side to \(30^{\circ}\) is \(y\). So \(\cos30^{\circ}=\frac{y}{12\sqrt{3}}\). Since \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\), we have \(\frac{\sqrt{3}}{2}=\frac{y}{12\sqrt{3}}\). Cross - multiply: \(y=\frac{12\sqrt{3}\times\sqrt{3}}{2}\). Simplify \(y=\frac{12\times3}{2}=18\).

Answer:

\(x = 6\sqrt{3}\), \(y = 18\)