Question

y = 8√3
color light purple
y =
e
27
60°
y
x =
number
x =
y =
color
y
h

Explanation:

Step1: Use trigonometric ratios

In a right - triangle, if one of the non - right angles is \(60^{\circ}\), then the other non - right angle is \(30^{\circ}\). We know that \(\sin60^{\circ}=\frac{27}{y}\) and \(\tan60^{\circ}=\frac{27}{x}\).
Since \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\) and \(\tan\theta = \frac{\text{opposite}}{\text{adjacent}}\) for a right - triangle. Also, \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\) and \(\tan60^{\circ}=\sqrt{3}\).

Step2: Solve for \(y\)

From \(\sin60^{\circ}=\frac{27}{y}\), we have \(y=\frac{27}{\sin60^{\circ}}\).
Substituting \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\), we get \(y = \frac{27}{\frac{\sqrt{3}}{2}}=18\sqrt{3}\).

Step3: Solve for \(x\)

From \(\tan60^{\circ}=\frac{27}{x}\), we have \(x=\frac{27}{\tan60^{\circ}}\).
Substituting \(\tan60^{\circ}=\sqrt{3}\), we get \(x=\frac{27}{\sqrt{3}} = 9\sqrt{3}\).

Answer:

\(x = 9\sqrt{3}\), \(y=18\sqrt{3}\)