Question

given right triangle xyz, what is the value of tan(y)?
options:

• $\frac{1}{2}$
• $\frac{sqrt{3}}{3}$
• $\frac{sqrt{3}}{2}$
• $\frac{2sqrt{3}}{3}$

Explanation:

Step1: Recall tangent definition

For angle $Y$, $\tan(Y)=\frac{\text{opposite side}}{\text{adjacent side}}$

Step2: Identify sides for $\angle Y$

In $\triangle XYZ$, right-angled at $Z$:

• Hypotenuse $XY=4$
• Opposite to $\angle Y$: $XZ$
• Adjacent to $\angle Y$: $YZ$

Step3: Calculate $XZ$ (opposite to $\angle Y$)

Use $\sin(30^\circ)=\frac{XZ}{XY}$
$\sin(30^\circ)=\frac{1}{2}$, so $XZ=XY\times\sin(30^\circ)=4\times\frac{1}{2}=2$

Step4: Calculate $YZ$ (adjacent to $\angle Y$)

Use $\cos(30^\circ)=\frac{YZ}{XY}$
$\cos(30^\circ)=\frac{\sqrt{3}}{2}$, so $YZ=4\times\frac{\sqrt{3}}{2}=2\sqrt{3}$

Step5: Compute $\tan(Y)$

$\tan(Y)=\frac{XZ}{YZ}=\frac{2}{2\sqrt{3}}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}$

Answer:

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