Question

given right triangle xyz, what is the value of tan(y)?
options:
\\(\frac{1}{2}\\)
\\(\frac{\sqrt{3}}{3}\\)
\\(\frac{\sqrt{3}}{2}\\)
\\(\frac{2\sqrt{3}}{3}\\)
(image of right triangle xyz with right angle at z, angle at x is 60°, angle at y is 30°, and side xy is 4)

Explanation:

Step1: Recall the definition of tangent in a right - triangle

In a right - triangle \( \triangle XYZ\) with right - angle at \(Z\), \(\tan(Y)=\frac{\text{opposite}}{\text{adjacent}}\). The side opposite to angle \(Y\) is \(XZ\) and the side adjacent to angle \(Y\) is \(YZ\).

Step2: Use the properties of a \(30 - 60-90\) triangle

In a \(30 - 60 - 90\) triangle, if the hypotenuse \(XY = 4\), then the side opposite the \(30^{\circ}\) angle (\(XZ\)) is \(\frac{1}{2}\) of the hypotenuse. So \(XZ=\frac{1}{2}\times4 = 2\).
The side opposite the \(60^{\circ}\) angle (\(YZ\)) is \(2\sqrt{3}\) (using the ratio \(1:\sqrt{3}:2\) for \(30 - 60 - 90\) triangle where sides are \(a,a\sqrt{3},2a\) and \(a = 2\)).

Step3: Calculate \(\tan(Y)\)

\(\tan(Y)=\frac{XZ}{YZ}\). Substituting \(XZ = 2\) and \(YZ=2\sqrt{3}\), we get \(\tan(Y)=\frac{2}{2\sqrt{3}}=\frac{1}{\sqrt{3}}\). Rationalizing the denominator (multiply numerator and denominator by \(\sqrt{3}\)), we have \(\tan(Y)=\frac{\sqrt{3}}{3}\)

Answer:

\(\frac{\sqrt{3}}{3}\) (the second option)