Question

what is the cosine ratio of ∠z? a. \\(\frac{3}{5}\\) b. \\(\frac{3}{4}\\) c. \\(\frac{4}{5}\\) d. \\(\frac{5}{3}\\) (with a right triangle xyz, right-angled at y, xy = 16, yz = 12)

Explanation:

Step1: Find the hypotenuse XZ

In right triangle \( XYZ \) with \( \angle Y = 90^\circ \), \( XY = 16 \), \( YZ = 12 \). By Pythagorean theorem, \( XZ=\sqrt{XY^{2}+YZ^{2}}=\sqrt{16^{2}+12^{2}}=\sqrt{256 + 144}=\sqrt{400} = 20 \).

Step2: Recall the cosine ratio

The cosine of an angle in a right triangle is the adjacent side over the hypotenuse. For \( \angle Z \), the adjacent side is \( YZ = 12 \) and the hypotenuse is \( XZ=20 \). So \( \cos Z=\frac{YZ}{XZ}=\frac{12}{20}=\frac{3}{5} \).

Answer:

A. \(\frac{3}{5}\)