Question

given right triangle gyk, what is the value of tan(g)?

1/2
√3/2
2√3/3
√3

Explanation:

Step1: Recall the tangent formula

In a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).

Step2: Identify the sides relative to angle \(G\)

For angle \(G\) in right - triangle \(GYK\), the opposite side to \(G\) is \(YK\) and the adjacent side to \(G\) is \(GK = 27\).
Since \(\angle G=60^{\circ}\), we know that \(\tan(G)=\tan(60^{\circ})\).

Step3: Calculate \(\tan(60^{\circ})\)

We know from the unit circle or trigonometric ratios of special angles that \(\tan(60^{\circ})=\sqrt{3}\).
Another way: In a \(30 - 60-90\) triangle, if the side adjacent to the \(60^{\circ}\) angle (\(GK\)) is \(x = 27\), the side opposite to the \(60^{\circ}\) angle (\(YK\)) is \(x\sqrt{3}\). Then \(\tan(G)=\frac{YK}{GK}=\frac{27\sqrt{3}}{27}=\sqrt{3}\).

Answer:

D. \(\sqrt{3}\)