Question

given right triangle xyz, what is the value of tan(y)?
○ \\(\frac{1}{2}\\)
○ \\(\frac{sqrt{3}}{3}\\)
○ \\(\frac{sqrt{3}}{2}\\)
○ \\(\frac{2sqrt{3}}{3}\\)

Explanation:

Step1: Recall tangent definition

For an acute angle in a right triangle, $\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}$. For $\angle Y$, opposite side is $XZ$, adjacent side is $YZ$.

Step2: Identify triangle sides

Given hypotenuse $XY=4$, $\angle Y=30^\circ$.

• Opposite to $\angle Y$: $XZ = XY \cdot \sin(30^\circ) = 4 \cdot \frac{1}{2} = 2$
• Adjacent to $\angle Y$: $YZ = XY \cdot \cos(30^\circ) = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}$

Step3: Calculate $\tan(Y)$

Substitute values into tangent formula:
$\tan(Y) = \frac{XZ}{YZ} = \frac{2}{2\sqrt{3}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$

Answer:

$\boldsymbol{\frac{\sqrt{3}}{3}}$ (corresponding to the second option)