To determine the value of [tex]\(\tan 60^\circ\)[/tex], let's start with the definition of the tangent function. The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.
For the angle [tex]\(60^\circ\)[/tex], we can use the properties of a 30-60-90 triangle, which has side ratios of 1:√3:2.
- The side opposite [tex]\(60^\circ\)[/tex] is [tex]\(\sqrt{3}\)[/tex]
- The side adjacent to [tex]\(60^\circ\)[/tex] is 1.
Thus, the tangent of [tex]\(60^\circ\)[/tex] is given by the ratio of the opposite side to the adjacent side:
[tex]\[
\tan 60^\circ = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{3}}{1} = \sqrt{3}
\][/tex]
So, the correct answer is:
C. [tex]\(\sqrt{3}\)[/tex].
To verify the numerical value, the tangent of [tex]\(60^\circ\)[/tex] approximately equals [tex]\(1.7320508075688767\)[/tex], which matches [tex]\(\sqrt{3}\)[/tex]. This confirms that our detailed stepwise explanation and conclusion are correct.
