Question

given right triangle xyz, what is the value of tan(60°)?

Explanation:

Step1: Recall tangent function definition

The tangent of an angle in a right - triangle is defined as $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. In right - triangle $XYZ$ with $\theta = 60^{\circ}$, the side opposite to the $60^{\circ}$ angle is $YZ$ and the side adjacent to the $60^{\circ}$ angle is $XZ$.

Step2: Identify sides for $\theta = 60^{\circ}$

If we consider $\angle X=60^{\circ}$, the side opposite to $\angle X$ is $YZ$ and the side adjacent to $\angle X$ is $XZ$. Given $XZ = 21$ and using the Pythagorean theorem or the properties of a 30 - 60 - 90 triangle, if the shorter leg (adjacent to the $60^{\circ}$ angle) is $a = 21$, the longer leg (opposite to the $60^{\circ}$ angle) $YZ=21\sqrt{3}$. Then $\tan(60^{\circ})=\frac{YZ}{XZ}$.

Step3: Calculate the value of $\tan(60^{\circ})$

Substitute $YZ = 21\sqrt{3}$ and $XZ = 21$ into the tangent formula: $\tan(60^{\circ})=\frac{21\sqrt{3}}{21}=\sqrt{3}$.

Answer:

$\sqrt{3}$