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What is the value of [tex]\tan \left(60^{\circ}\right)[/tex]?

A. [tex]\frac{1}{2}[/tex]
B. [tex]\sqrt{3}[/tex]
C. [tex]\frac{\sqrt{3}}{2}[/tex]
D. [tex]\frac{1}{\sqrt{3}}[/tex]


Sagot :

To determine the value of [tex]\(\tan(60^\circ)\)[/tex], let's start by understanding the relationship between angles and their tangent values.

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. Specifically, for [tex]\(\tan(60^\circ)\)[/tex]:

[tex]\[ \tan(60^\circ) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]

In a 30-60-90 right triangle, the ratios of the sides are known:
- The side opposite the 30° angle is [tex]\(a\)[/tex],
- The side opposite the 60° angle is [tex]\(a\sqrt{3}\)[/tex],
- The hypotenuse is [tex]\(2a\)[/tex].

This results in:

[tex]\[ \tan(60^\circ) = \frac{\text{opposite side of } 60^\circ}{\text{adjacent side of } 60^\circ} = \frac{a\sqrt{3}}{a} = \sqrt{3} \][/tex]

Hence, the value of [tex]\(\tan(60^\circ)\)[/tex] is [tex]\(\sqrt{3}\)[/tex].

The correct answer is:

[tex]\(\sqrt{3}\)[/tex]