Question

from unit 4, lesson 3
here is triangle abc.
what is the length of side ab?

Explanation:

Step1: Recall the tangent function

In a right - triangle, the tangent of an angle is defined as \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For \(\angle C = 30^{\circ}\) in right - triangle \(ABC\) with right - angle at \(A\), \(\tan C=\tan30^{\circ}=\frac{AB}{AC}\).

Step2: Substitute the known values

We know that \(AC = 6\) and \(\tan30^{\circ}=\frac{\sqrt{3}}{3}\). Substituting into the formula \(\tan30^{\circ}=\frac{AB}{6}\).

Step3: Solve for \(AB\)

Cross - multiply: \(AB = 6\times\tan30^{\circ}\). Since \(\tan30^{\circ}=\frac{\sqrt{3}}{3}\), then \(AB = 6\times\frac{\sqrt{3}}{3}\). Simplify the expression: \(AB = 2\sqrt{3}\).

Answer:

\(2\sqrt{3}\)