Question

c
color clark blue
d
8
x=
number
x=
y=
color
y=

Explanation:

Step1: Use trigonometric ratios

In a right - triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\) and \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Given \(\theta = 30^{\circ}\), the side opposite to \(30^{\circ}\) is \(8\), the hypotenuse is \(x\), and the adjacent side is \(y\).
We know that \(\sin30^{\circ}=\frac{8}{x}\). Since \(\sin30^{\circ}=\frac{1}{2}\), we have \(\frac{1}{2}=\frac{8}{x}\).

Step2: Solve for \(x\)

Cross - multiply: \(x = 16\).

Step3: Solve for \(y\)

Using the Pythagorean theorem \(x^{2}=y^{2}+8^{2}\) (or \(\tan30^{\circ}=\frac{8}{y}\), since \(\tan30^{\circ}=\frac{\sqrt{3}}{3}\), then \(y = 8\sqrt{3}\)).
By Pythagorean theorem: \(y=\sqrt{x^{2}-8^{2}}=\sqrt{16^{2}-8^{2}}=\sqrt{(16 + 8)(16 - 8)}=\sqrt{24\times8}=\sqrt{192}=8\sqrt{3}\)

Answer:

\(x = 16\), \(y = 8\sqrt{3}\)