Question

find the exact value of tan 60°.
tan 60° =
(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 \(60^\circ\) angle in an equilateral triangle (which can be split into two \(30 - 60 - 90\) right triangles), we know the side ratios. In a \(30 - 60 - 90\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\) (opposite \(30^\circ\), opposite \(60^\circ\), hypotenuse respectively). So, for an angle of \(60^\circ\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For \(60^\circ\), the opposite side is \(\sqrt{3}\) and the adjacent side is \(1\) (in the \(30 - 60 - 90\) triangle with side lengths relative to \(30^\circ\) as \(1\), \(60^\circ\) as \(\sqrt{3}\), hypotenuse \(2\)).

Step2: Calculate \(\tan 60^\circ\)

Using the definition \(\tan\theta = \frac{\sin\theta}{\cos\theta}\), we know that \(\sin 60^\circ=\frac{\sqrt{3}}{2}\) and \(\cos 60^\circ=\frac{1}{2}\). Then \(\tan 60^\circ=\frac{\sin 60^\circ}{\cos 60^\circ}=\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}\). When dividing by a fraction, we multiply by its reciprocal, so \(\frac{\sqrt{3}}{2}\times\frac{2}{1}=\sqrt{3}\).

Answer:

\(\sqrt{3}\)