To find the value of [tex]\(\tan 60^\circ\)[/tex], we first recall some properties and values from trigonometry, specifically the tangent of common angles.
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 some special angles like [tex]\(30^\circ\)[/tex], [tex]\(45^\circ\)[/tex], and [tex]\(60^\circ\)[/tex], we have well-known values for the trigonometric functions.
For a [tex]\(60^\circ\)[/tex] angle, we typically refer to the 30-60-90 special right triangle. In this triangle:
- The side opposite the [tex]\(30^\circ\)[/tex] angle (the shorter leg) has a length of [tex]\(1\)[/tex].
- The side opposite the [tex]\(60^\circ\)[/tex] angle (the longer leg) has a length of [tex]\(\sqrt{3}\)[/tex].
- The hypotenuse has a length of [tex]\(2\)[/tex].
The tangent of [tex]\(60^\circ\)[/tex] is given by the ratio of the length of the side opposite the [tex]\(60^\circ\)[/tex] angle to the length of the side adjacent to it. Therefore:
[tex]\[
\tan 60^\circ = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{\sqrt{3}}{1} = \sqrt{3}
\][/tex]
Thus, the value of [tex]\(\tan 60^\circ\)[/tex] is [tex]\(\sqrt{3}\)[/tex].
Checking the multiple-choice answers provided:
A. [tex]\(\frac{1}{2}\)[/tex]
B. [tex]\(\frac{2}{\sqrt{3}}\)[/tex]
C. [tex]\(\frac{\sqrt{3}}{2}\)[/tex]
D. [tex]\(\sqrt{3}\)[/tex]
E. [tex]\(\frac{1}{\sqrt{3}}\)[/tex]
F. 1
The correct answer that matches [tex]\(\sqrt{3}\)[/tex] is:
D. [tex]\(\sqrt{3}\)[/tex]
