What is the value of [tex]\(\tan(60^\circ)\)[/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]



Answer :

To find the value of [tex]\(\tan(60^\circ)\)[/tex], follow these steps:

1. Identify the Angle: We are dealing with an angle of [tex]\(60^\circ\)[/tex].

2. 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 the angle [tex]\(60^\circ\)[/tex], which is one of the special angles in trigonometry, this ratio is well-known.

3. Use the Known Values: For the special angles, particularly in a 30-60-90 triangle, the sides have a specific ratio:
- The side opposite the [tex]\(30^\circ\)[/tex] angle has a length of [tex]\(1\)[/tex].
- The side opposite the [tex]\(60^\circ\)[/tex] angle (which we need) has a length of [tex]\(\sqrt{3}\)[/tex].
- The hypotenuse has a length of [tex]\(2\)[/tex].

4. Calculate Tangent: Using the definition of tangent:
[tex]\[ \tan(60^\circ) = \frac{\text{length of the opposite side}}{\text{length of the adjacent side}} = \frac{\sqrt{3}}{1} = \sqrt{3} \][/tex]

5. Confirm by comparing with standard trigonometric values:
- We know that [tex]\(\tan(60^\circ) = \sqrt{3}\)[/tex].

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