Question

alue of each variable.
11.

x =

y =

Explanation:

Step1: Identify the triangle type

This is a right - triangle with one angle \(60^{\circ}\), so the other non - right angle is \(30^{\circ}\) (since the sum of angles in a triangle is \(180^{\circ}\) and one angle is \(90^{\circ}\)). The hypotenuse is given as \(16\).

Step2: Find the value of \(x\) (opposite to \(60^{\circ}\) angle)

In a \(30 - 60 - 90\) triangle, the side opposite to \(60^{\circ}\) (let's call it \(a\)), the side opposite to \(30^{\circ}\) (let's call it \(b\)) and the hypotenuse (\(c\)) have the ratio \(a:b:c=\sqrt{3}:1:2\).
The side opposite to \(60^{\circ}\) ( \(x\)) is related to the hypotenuse (\(c = 16\)) by the formula \(\sin(60^{\circ})=\frac{x}{16}\). Since \(\sin(60^{\circ})=\frac{\sqrt{3}}{2}\), we have \(x = 16\times\sin(60^{\circ})=16\times\frac{\sqrt{3}}{2}=8\sqrt{3}\).

Step3: Find the value of \(y\) (adjacent to \(60^{\circ}\) angle)

We can use the formula \(\cos(60^{\circ})=\frac{y}{16}\). Since \(\cos(60^{\circ})=\frac{1}{2}\), we have \(y = 16\times\cos(60^{\circ})=16\times\frac{1}{2}=8\). Also, in a \(30 - 60 - 90\) triangle, the side opposite to \(30^{\circ}\) (which is \(y\)) is half of the hypotenuse. Since the hypotenuse is \(16\), \(y=\frac{16}{2}=8\).

Answer:

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