Question

find the exact value of tan 30°.
tan 30° =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall the definition of tangent

The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. For a \(30^\circ - 60^\circ - 90^\circ\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\) (opposite \(30^\circ\), opposite \(60^\circ\), hypotenuse respectively).
So, for \(30^\circ\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For \(30^\circ\), opposite side length is \(1\), adjacent side length is \(\sqrt{3}\).

Step2: Calculate \(\tan30^\circ\)

Using the definition \(\tan30^\circ=\frac{\sin30^\circ}{\cos30^\circ}\). We know that \(\sin30^\circ = \frac{1}{2}\) and \(\cos30^\circ=\frac{\sqrt{3}}{2}\). Then \(\tan30^\circ=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}\). When dividing fractions, we multiply by the reciprocal, so \(\frac{1}{2}\times\frac{2}{\sqrt{3}}=\frac{1}{\sqrt{3}}\). Rationalizing the denominator, we multiply numerator and denominator by \(\sqrt{3}\), getting \(\frac{\sqrt{3}}{3}\).

Answer:

\(\frac{\sqrt{3}}{3}\)