Question

determining a tangent ratio
what is the value of \\(\tan(60^\circ)\\)?
\\(\circ\\) \\(\frac{1}{2}\\)
\\(\circ\\) \\(\sqrt{3}\\)
\\(\circ\\) \\(\frac{\sqrt{3}}{2}\\)
\\(\circ\\) \\(\frac{1}{\sqrt{3}}\\)

Explanation:

Step1: Recall tangent definition

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Step2: Identify sides for $60^\circ$

For $\angle B = 60^\circ$, opposite side $AC$, adjacent side $BC=6$. First calculate $AC$ via Pythagoras:
$AC = \sqrt{AB^2 - BC^2} = \sqrt{12^2 - 6^2} = \sqrt{144-36} = \sqrt{108} = 6\sqrt{3}$

Step3: Compute $\tan(60^\circ)$

$\tan(60^\circ) = \frac{AC}{BC} = \frac{6\sqrt{3}}{6} = \sqrt{3}$

Answer:

$\boldsymbol{\sqrt{3}}$ (Option: $\boldsymbol{\sqrt{3}}$)