Question

given right triangle gyk, what is the value of tan (g)? options: 1/3, √3, (2√3)/3, √3/3 (diagram: right triangle gyk with right angle at k, gk = 27, ∠g = 60°, ∠y = 30°)

Explanation:

Step1: Recall tangent definition

In a right triangle, \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\) to angle \(\theta\). For \(\angle G\) (which is \(60^\circ\)), we need to find the lengths of the opposite and adjacent sides.

Step2: Identify sides for \(\angle G\)

• Adjacent side to \(\angle G\) is \(GK = 27\).
• Let's find the opposite side ( \(YK\) ). In a \(30^\circ - 60^\circ - 90^\circ\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\) (short leg : long leg : hypotenuse). Here, \(GK\) is the short leg (opposite \(30^\circ\) angle at \(Y\)), so the long leg (opposite \(60^\circ\) angle, which is \(YK\)) is \(GK\times\sqrt{3}=27\sqrt{3}\).

Step3: Calculate \(\tan(G)\)

\(\tan(G)=\frac{\text{opposite to }G}{\text{adjacent to }G}=\frac{YK}{GK}=\frac{27\sqrt{3}}{27}=\sqrt{3}\)

Answer:

\(\sqrt{3}\)