Question

what is the measure in degrees of \\( \angle xyz \\) in the figure shown?

Explanation:

Step1: Recall the cosine formula

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Let \(\theta = \angle XYZ\), the adjacent side to \(\angle XYZ\) is \(XY = 4\sqrt{3}\), and the hypotenuse \(YZ=8\sqrt{3}\).

Step2: Calculate \(\cos\angle XYZ\)

\(\cos\angle XYZ=\frac{XY}{YZ}=\frac{4\sqrt{3}}{8\sqrt{3}}\).
Simplify the fraction: \(\frac{4\sqrt{3}}{8\sqrt{3}}=\frac{4}{8}=\frac{1}{2}\).

Step3: Find the angle whose cosine is \(\frac{1}{2}\)

We know that if \(\cos\theta=\frac{1}{2}\), and \(\theta\in[0^{\circ},180^{\circ}]\), then \(\theta = 60^{\circ}\) (since \(\cos60^{\circ}=\frac{1}{2}\)).

Answer:

\(60^{\circ}\)