Question

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

Explanation:

Step1: Identify sides relative to ∠Y

In right triangle \( XYZ \) (right - angled at \( Z \)), for angle \( Y = 30^{\circ}\), we use the definition of tangent: \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \). First, we find the lengths of the sides opposite and adjacent to \( \angle Y \).

In a 30 - 60 - 90 triangle, the hypotenuse \( XY = 4 \). The side opposite \( 30^{\circ}\) (which is \( XZ \)) is half the hypotenuse, so \( XZ=\frac{1}{2}\times XY=\frac{1}{2}\times4 = 2 \). The side adjacent to \( 30^{\circ}\) (which is \( YZ \)) can be found using the Pythagorean theorem or the ratio of sides in a 30 - 60 - 90 triangle. The ratio of sides in a 30 - 60 - 90 triangle is \( 1:\sqrt{3}:2 \) (opposite 30° : opposite 60° : hypotenuse). So the side opposite \( 60^{\circ}\) (which is \( YZ \)) is \( \sqrt{3}\times \) (side opposite 30°). Since the side opposite 30° is 2, \( YZ = 2\sqrt{3} \).

Step2: Calculate \( \tan(Y) \)

For \( \angle Y \), the opposite side is \( XZ = 2 \) and the adjacent side is \( YZ=2\sqrt{3} \). Then \( \tan(Y)=\frac{\text{opposite to }Y}{\text{adjacent to }Y}=\frac{XZ}{YZ}=\frac{2}{2\sqrt{3}} \).

Simplify \( \frac{2}{2\sqrt{3}}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3} \) (rationalizing the denominator: multiply numerator and denominator by \( \sqrt{3} \), \( \frac{1\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}}=\frac{\sqrt{3}}{3} \)).

Answer:

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