Question

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

Explanation:

Step1: Recall tangent definition

For an angle in a right triangle, $\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}$. For $\angle G$, opposite side is $YK$, adjacent side is $GK=27$.

Step2: Find length of $YK$

Use $\tan(60^\circ) = \frac{YK}{GK}$. We know $\tan(60^\circ)=\sqrt{3}$, so:
$$YK = GK \times \tan(60^\circ) = 27 \times \sqrt{3}$$

Step3: Calculate $\tan(G)$

Substitute into tangent formula:
$$\tan(G) = \frac{YK}{GK} = \frac{27\sqrt{3}}{27} = \sqrt{3}$$

Answer:

$\sqrt{3}$ (Option D)